Case Study / Scenario-Based MCQs for Sub-Topics of Topic 5: Construction

Question 1. Amit needs to construct a circle of radius $4 \text{ cm}$. He opens his compass such that the distance between the sharp point and the pencil tip is $4 \text{ cm}$. He places the sharp point at the desired location for the centre and rotates the compass to draw the circle. Which tool(s) are essential for this task as described?

(A) Only a compass

(B) A compass and a ruler (with markings)

(C) A compass and a protractor

(D) A ruler and a protractor

Question 2. Amit now needs to construct a line segment of length $7 \text{ cm}$. He takes his ruler marked in millimetres. To ensure accuracy, where should he start marking the segment on the paper using the ruler?

(A) At the $7 \text{ cm}$ mark, drawing backwards to the $0 \text{ cm}$ mark.

(B) At the $0 \text{ cm}$ mark, drawing forward to the $7 \text{ cm}$ mark.

(C) At any convenient mark (say $2 \text{ cm}$), and draw to the mark that is $7 \text{ cm}$ away (i.e., $9 \text{ cm}$).

(D) Options (B) and (C) are both valid methods for accurate construction.

Question 3. Amit has an existing line segment PQ. He needs to copy this segment onto a line L, starting from a point R on L. Which sequence of actions using a compass and ruler correctly performs this task?

(A) Measure the length of PQ with a ruler, then mark a segment of that length on L starting from R using the ruler.

(B) Set the compass opening to the length of PQ. Place the compass point at R. Draw an arc intersecting L at S. RS is the copied segment.

(C) Place the compass point at P and the pencil at Q. Without changing the opening, place the compass point at R and draw a line through R.

(D) Draw a circle centered at R with a radius equal to the length of PQ.

Question 4. Amit constructs a circle with a diameter of $10 \text{ cm}$. What is true about the distance from the center of this circle to any point on its circumference?

(A) It is $10 \text{ cm}$.

(B) It is $5 \text{ cm}$.

(C) It is $\pi \times 10 \text{ cm}$.

(D) It varies depending on the point on the circumference.

Question 5. Amit is using a ruler with markings in centimetres and millimetres. He needs to mark points for a line segment. What is the smallest unit of length he can accurately mark and measure using this ruler?

(A) $1 \text{ cm}$

(B) $0.5 \text{ cm}$

(C) $1 \text{ mm}$

(D) $0.1 \text{ mm}$

Question 6. Amit needs to draw several line segments of the exact same length as a given segment AB, but at different positions on the paper. Which is the most efficient and accurate method for him to reproduce the length of AB multiple times?

(A) Measure AB with a ruler and carefully use the ruler to draw segments of that length each time.

(B) Set the compass opening to the length of AB once, and use the compass to mark the endpoints of each new segment on a line drawn with a ruler.

(C) Use a protractor to get the correct angle and draw the segment.

(D) Visually estimate the length and draw the segments.

Question 1. Priya needs to construct a $60^\circ$ angle using compass and ruler. She draws a ray OA. With O as the center, she draws an arc of a convenient radius intersecting OA at B. What is her next step to construct the $60^\circ$ angle?

(A) With A as the center and the same radius, draw an arc.

(B) With B as the center and the same radius, draw an arc intersecting the first arc at C.

(C) With O as the center and a larger radius, draw another arc.

(D) Measure $60^\circ$ with a protractor and draw the angle.

Question 2. Priya successfully constructs a $60^\circ$ angle. Now she needs to bisect it. With the vertex as center, she draws an arc intersecting both arms. Let the intersection points be P and Q. What should be her next step to draw the angle bisector?

(A) Draw a line segment PQ.

(B) With P as center, draw an arc in the interior of the angle. With Q as center and a *different* radius, draw another arc.

(C) With P as center, draw an arc in the interior of the angle. With Q as center and the *same* radius, draw another arc intersecting the first arc at R.

(D) Measure the angle with a protractor and divide the reading by two.

Question 3. Priya constructs a $90^\circ$ angle at a point M on a line. She draws arcs of the same radius from M intersecting the line at A and B. Then from A and B, with a radius greater than AM, she draws arcs intersecting at N. She joins M to N. Which property justifies that $\angle AMN = 90^\circ$?

(A) Triangle AMN is isosceles.

(B) Triangle AQN is equilateral where Q is the intersection of arcs from A and B.

(C) MN is the perpendicular bisector of AB (where AB is the segment on the line centered at M), and the line AB is a straight angle ($180^\circ$).

(D) Angles on a straight line sum to $180^\circ$.

Question 4. Priya wants to construct a $75^\circ$ angle. She knows she can combine standard constructions. Which combination of standard constructible angles can give her a $75^\circ$ angle?

(A) $30^\circ + 45^\circ$

(B) $90^\circ - 15^\circ$

(C) $60^\circ + 15^\circ$

(D) Bisecting a $150^\circ$ angle

Question 5. Priya has constructed an angle bisector of $\angle ABC$. She claims that any point P on the bisector is equidistant from ray BA and ray BC. Which geometric concept supports Priya's claim?

(A) The definition of an angle bisector.

(B) The property of angles on a straight line.

(C) Triangle congruence, often used to prove this property.

(D) The property that points on the bisector are equidistant from the vertex B.

Question 6. Priya is asked to construct an angle of $40^\circ$. Based on standard compass and ruler constructions, can she accurately construct this angle?

(A) Yes, by dividing a $120^\circ$ angle into three equal parts.

(B) Yes, by combining $30^\circ$ and $10^\circ$ angles.

(C) No, $40^\circ$ is not an angle that can be constructed using only compass and ruler by standard Euclidean methods.

(D) Yes, by using a protractor first and then copying the angle with compass and ruler.

Question 1. Rohan needs to construct a perpendicular to a line XY at a point P on XY. He draws arcs of the same radius from P, intersecting XY at A and B. Then, with A and B as centers, he draws arcs with a radius greater than AP, intersecting at Q. He joins P to Q. What angle does the line PQ make with the line XY?

(A) $60^\circ$

(B) $45^\circ$

(C) $90^\circ$

(D) $120^\circ$

Question 2. Rohan has a line 'l' and a point R not on 'l'. He wants to construct a perpendicular from R to 'l'. He draws an arc from R intersecting 'l' at points C and D. What is his next step?

(A) Find the midpoint of CD.

(B) From C and D, draw arcs with the same radius on the opposite side of 'l' from R, intersecting at S.

(C) Join R to the midpoint of CD.

(D) Draw a line through R parallel to 'l'.

Question 3. Rohan needs to find the midpoint of a line segment AB. Which construction should he perform?

(A) Construct the angle bisector of the angle formed by AB and a ray from A.

(B) Construct the perpendicular bisector of AB.

(C) Construct a perpendicular line from A to AB.

(D) Measure the length of AB and divide by two using a ruler.

Question 4. Rohan constructs the perpendicular bisector of segment AB. He claims that any point P on this line is equidistant from A and B (i.e., PA = PB). Which property justifies this claim?

(A) Any point on the perpendicular bisector is equidistant from the endpoints of the segment.

(B) The perpendicular bisector forms a $90^\circ$ angle with the segment.

(C) The perpendicular bisector passes through the midpoint.

(D) Triangle PAB is always a right-angled triangle.

Question 5. Rohan needs to draw a perpendicular to a line from a point outside the line. The point represents his house, and the line represents a road. He wants to find the point on the road closest to his house. Which geometric principle states that the shortest distance from a point to a line is along the perpendicular segment?

(A) Triangle Inequality Theorem.

(B) Pythagorean Theorem.

(C) The perpendicular distance is the shortest distance.

(D) Distance formula in coordinate geometry.

Question 6. Rohan constructs the perpendicular bisector of segment AB. This construction creates two points P and Q by the intersection of arcs from A and B. The line PQ is the perpendicular bisector. Which geometric figure is formed by points A, B, P, and Q?

(A) Square

(B) Rectangle

(C) Rhombus

(D) Trapezium

Question 1. Sanjay is given a line 'l' and a point P not on 'l'. He wants to construct a line through P parallel to 'l' using the corresponding angles method. He draws a transversal line through P intersecting 'l' at Q. He identifies a corresponding angle at Q. What is the crucial step he needs to perform at point P to construct the parallel line?

(A) Construct a $90^\circ$ angle at P.

(B) Construct a line through P that is perpendicular to the transversal.

(C) Copy the corresponding angle from Q to P in the corresponding position relative to the transversal and P.

(D) Copy the alternate interior angle from Q to P.

Question 2. Sanjay successfully copies the corresponding angle at P. The ray forming this copied angle will be part of the parallel line. Which geometric theorem or postulate guarantees that the line formed by this ray is indeed parallel to line 'l'?

(A) Alternate Interior Angles Theorem.

(B) Corresponding Angles Postulate (or its converse).

(C) Angle Sum Property of a triangle.

(D) Vertically Opposite Angles Theorem.

Question 3. Sanjay could also use the alternate interior angles method. He draws a transversal through P intersecting 'l' at Q. He identifies an alternate interior angle at Q. Where should he copy this angle at point P?

(A) On the same side of the transversal as the angle at Q, between the lines.

(B) On the opposite side of the transversal as the angle at Q, between the lines.

(C) On the corresponding side of the transversal.

(D) On the vertically opposite side of the transversal at P.

Question 4. Sanjay constructs a line m through P parallel to line 'l'. He claims that any line that intersects 'l' must also intersect line m, unless it is parallel to both.

This statement is related to the definition and properties of parallel lines. Under standard Euclidean geometry, how many lines can be drawn through P that are parallel to 'l'?

(A) Zero

(B) Exactly one

(C) Two

(D) Infinite

Question 5. Sanjay's teacher asks him to justify his parallel line construction using angle copying. The justification relies on the converses of angle theorems. What is the converse of the statement: "If two lines are parallel, then corresponding angles are equal"?

(A) If corresponding angles are equal, then the two lines are parallel.

(B) If two lines are parallel, then alternate interior angles are equal.

(C) If corresponding angles are not equal, then the lines are not parallel.

(D) If two lines are not parallel, then corresponding angles are not equal.

Question 6. Sanjay uses his compass and ruler to copy an angle. He draws an arc from the vertex intersecting the arms. He then draws a similar arc from the new vertex. What does he use the compass to measure next?

(A) The angle measure in degrees.

(B) The length of the arc segment.

(C) The straight-line distance between the points where the first arc intersected the arms of the original angle.

(D) The distance from the vertex to the intersection point on one arm.

Question 1. Nisha has a line segment AB of length $10 \text{ cm}$. She needs to divide it internally in the ratio $2:3$. She draws a ray AC from A, making an acute angle with AB. How many equal parts should she mark on the ray AC using her compass?

(A) 2

(B) 3

(C) 5

(D) 10

Question 2. Nisha marks 5 equal points on ray AC as $A_1, A_2, A_3, A_4, A_5$. Which point on ray AC should she connect to point B on the line segment AB?

(A) $A_2$

(B) $A_3$

(C) $A_5$

(D) $A_1$

Question 3. Nisha connects $A_5$ to B. Now, to find the point P on AB that divides it in the ratio $2:3$, she needs to draw a line parallel to $A_5B$. Through which point on ray AC should she draw this parallel line?

(A) $A_2$ (since the ratio is 2:3)

(B) $A_3$ (since the ratio is 2:3)

(C) $A_1$

(D) $A_4$

Question 4. The point P found by Nisha's construction divides AB such that AP/PB = $2/3$. Which geometric theorem justifies this result?

(A) Pythagorean Theorem

(B) Angle Bisector Theorem

(C) Basic Proportionality Theorem (Thales Theorem)

(D) Midpoint Theorem

Question 5. Nisha needs to divide another line segment into 4 equal parts. Using the same construction method involving a ray, how many equal parts should she mark on the auxiliary ray?

(A) 1

(B) 2

(C) 4

(D) 8

Question 6. Nisha uses her compass to mark the equal segments on the ray AC. What property of the compass is essential for this step?

(A) It can draw circles.

(B) It can be opened to any desired radius.

(C) It can maintain a fixed distance between its points to transfer lengths accurately.

(D) It has a sharp point to mark the paper.

Question 1. Varun needs to construct a triangle with sides $5 \text{ cm}$, $6 \text{ cm}$, and $7 \text{ cm}$ using the SSS criterion. He draws the base BC of length $7 \text{ cm}$. What are the next steps he should take?

(A) Draw an arc from B with radius $5 \text{ cm}$ and an arc from C with radius $6 \text{ cm}$. The intersection is A.

(B) Draw an arc from B with radius $6 \text{ cm}$ and an arc from C with radius $5 \text{ cm}$. The intersection is A.

(C) Either (A) or (B) will result in the correct triangle.

(D) Check if $5+6 > 7$, $5+7 > 6$, and $6+7 > 5$ first.

Question 2. Varun is given sides of length $4 \text{ cm}$ and $5 \text{ cm}$ and an included angle of $50^\circ$ (SAS criterion) to construct a triangle. Which sequence of construction steps is correct?

(A) Draw a segment of length $4 \text{ cm}$. Construct the $50^\circ$ angle at one endpoint. Mark a segment of length $5 \text{ cm}$ on the arm of the angle. Join the endpoints.

(B) Draw a segment of length $4 \text{ cm}$. Construct the $50^\circ$ angle at one endpoint. Mark a segment of length $5 \text{ cm}$ from the *other* endpoint of the first segment.

(C) Draw a segment of length $5 \text{ cm}$. Construct the $50^\circ$ angle at one endpoint. Mark a segment of length $4 \text{ cm}$ on the arm of the angle. Join the endpoints.

(D) Both (A) and (C) are valid methods for constructing the same triangle (up to congruence).

Question 3. Varun is given angles $60^\circ$ and $40^\circ$ and the included side $6 \text{ cm}$ (ASA criterion) to construct a triangle. He draws the base AB of length $6 \text{ cm}$. What should he do next?

(A) Construct an angle of $60^\circ$ at A and an angle of $40^\circ$ at B on the same side of AB.

(B) Construct an angle of $60^\circ$ at A and an angle of $40^\circ$ at B on opposite sides of AB.

(C) Calculate the third angle ($180^\circ - 60^\circ - 40^\circ = 80^\circ$) and construct it at A.

(D) Draw arcs from A and B with radii related to the angles.

Question 4. Varun is given angles $50^\circ$ and $70^\circ$ and a side of $8 \text{ cm}$ opposite the $70^\circ$ angle (AAS criterion). How can he construct this triangle using standard compass and ruler methods?

(A) Draw side BC = $8 \text{ cm}$. Construct $\angle B=50^\circ$ at B and $\angle C=70^\circ$ at C.

(B) Calculate the third angle ($60^\circ$). Draw a side of length $8 \text{ cm}$ and construct angles of $50^\circ$ and $60^\circ$ at its endpoints on the same side.

(C) Draw a side of length $8 \text{ cm}$. Construct the $70^\circ$ angle opposite this side and the $50^\circ$ angle at one endpoint.

(D) Use the angle sum property to find the third angle and then use the ASA criterion with the given side and the two angles adjacent to it.

Question 5. Varun tries to construct a triangle with sides $4 \text{ cm}$, $5 \text{ cm}$, and $10 \text{ cm}$ using the SSS criterion. He draws a base of $10 \text{ cm}$. He then attempts to draw arcs from the endpoints with radii $4 \text{ cm}$ and $5 \text{ cm}$. What will happen?

(A) The arcs will intersect at two points.

(B) The arcs will be tangent to each other.

(C) The arcs will not intersect at all.

(D) The arcs will intersect at only one point.

Question 6. Varun is given two sides of a triangle and the measure of an angle that is NOT included between them (SSA case). For example, sides $5 \text{ cm}$, $7 \text{ cm}$ and an angle of $30^\circ$ opposite the $5 \text{ cm}$ side. Is a unique triangle always determined by this information?

(A) Yes, always.

(B) No, it is sometimes possible to construct two different triangles (the ambiguous case).

(C) No, a triangle can never be constructed with this information.

(D) Yes, but only if the given angle is $90^\circ$ (RHS case).

Question 1. Deepa needs to construct an equilateral triangle of side $5 \text{ cm}$. She draws a base of length $5 \text{ cm}$. What is the most direct way to find the third vertex using a compass?

(A) Construct $60^\circ$ angles at both ends of the base.

(B) Draw arcs of radius $5 \text{ cm}$ from both endpoints of the base, intersecting above the base.

(C) Construct the perpendicular bisector of the base.

(D) Measure the height of the equilateral triangle and mark it above the midpoint of the base.

Question 2. Deepa constructs an isosceles triangle with a base of $6 \text{ cm}$ and equal sides of $8 \text{ cm}$. Which triangle construction criterion did she use for this task?

(A) SSS

(B) SAS

(C) ASA

(D) RHS

Question 3. Deepa needs to construct a right-angled triangle with a hypotenuse of $13 \text{ cm}$ and one leg of $5 \text{ cm}$. She draws the leg of length $5 \text{ cm}$ as the base (say AB). At point A, she constructs a $90^\circ$ angle. What should she do next to locate the third vertex C?

(A) Draw a line from B perpendicular to the arm of the $90^\circ$ angle.

(B) Draw an arc from B with radius $13 \text{ cm}$, intersecting the arm of the $90^\circ$ angle.

(C) Draw an arc from A with radius $13 \text{ cm}$, intersecting the arm of the $90^\circ$ angle.

(D) Measure $12 \text{ cm}$ along the arm of the $90^\circ$ angle from A (since $5^2 + 12^2 = 13^2$).

Question 4. Deepa constructs an isosceles triangle with a base of $7 \text{ cm}$ and base angles of $50^\circ$ each. What is the measure of the vertex angle (the angle opposite the base) in this triangle?

(A) $50^\circ$

(B) $60^\circ$

(C) $80^\circ$

(D) $100^\circ$

Question 5. Deepa's teacher asks for the justification of constructing the equilateral triangle by drawing arcs of equal radius from the base endpoints. The justification is based on showing that the triangle formed has three equal sides. Which geometric principle is applied here?

(A) Angles opposite equal sides are equal.

(B) SSS congruence criterion.

(C) Properties of circles (all radii are equal).

(D) Angle sum property of a triangle.

Question 6. Deepa needs to construct an isosceles triangle given the length of the base and the measure of the vertex angle. For example, base $8 \text{ cm}$ and vertex angle $40^\circ$. What is a necessary first step after drawing the base?

(A) Construct the vertex angle at one endpoint of the base.

(B) Calculate the measure of the base angles using the angle sum property.

(C) Construct the perpendicular bisector of the base.

(D) Draw arcs from the endpoints of the base with the same radius.

Question 1. Kartik needs to construct $\triangle ABC$ given BC = $7 \text{ cm}$, $\angle B = 75^\circ$, and AB + AC = $12 \text{ cm}$. He draws BC, constructs $\angle B$. Along the arm of $\angle B$, he marks a point D such that BD = $12 \text{ cm}$. He joins D to C. What is the crucial next step to find vertex A?

(A) Construct the angle bisector of $\angle BDC$.

(B) Construct a line through C parallel to BD.

(C) Construct the perpendicular bisector of CD.

(D) Draw an arc from C with radius equal to AC.

Question 2. Kartik needs to construct $\triangle ABC$ given BC = $6 \text{ cm}$, $\angle B = 60^\circ$, and AB - AC = $2 \text{ cm}$ (assume AB > AC). He draws BC, constructs $\angle B$. Along the arm of $\angle B$, he marks a point D such that BD = $2 \text{ cm}$. He joins D to C. What is the crucial next step to find vertex A?

(A) Construct the perpendicular bisector of CD.

(B) Construct the angle bisector of $\angle BCD$.

(C) Draw a line through D parallel to AC.

(D) Draw an arc from C with radius equal to AC.

Question 3. Kartik needs to construct $\triangle ABC$ given $\angle B = 60^\circ$, $\angle C = 45^\circ$, and perimeter AB + BC + CA = $15 \text{ cm}$. He draws a line segment PQ = $15 \text{ cm}$. At P, he constructs an angle of $(1/2)\angle B = 30^\circ$. At Q, he constructs an angle of $(1/2)\angle C = 22.5^\circ$. Let the arms of these angles intersect at A. What are the next steps to find vertices B and C on PQ?

(A) Construct angle bisectors of $\angle APQ$ and $\angle AQP$.

(B) Construct perpendicular bisectors of AP and AQ. Their intersections with PQ are B and C.

(C) Join A to the midpoint of PQ to find B.

(D) Draw lines through A parallel to PQ.

Question 4. Kartik needs to construct a triangle given two sides and a median, say AB=$5 \text{ cm}$, AC=$6 \text{ cm}$, and median AD=$4 \text{ cm}$ to BC. He extends AD to E such that AD=DE and joins C to E. What type of quadrilateral is ABEC?

(A) Square

(B) Rhombus

(C) Rectangle

(D) Parallelogram

Question 5. In the construction of a triangle given two sides (AB, AC) and the median AD, after forming parallelogram ABEC, Kartik needs to construct triangle ACE. What are the lengths of the sides of triangle ACE?

(A) AB, AC, AD

(B) AB, AC, $2 \cdot \text{AD}$

(C) AC, BE, AE

(D) AC, AB, AE (where AE = 2*AD)

Question 6. Kartik is asked to construct a triangle given two angles and an altitude. For example, $\angle B = 60^\circ, \angle C = 45^\circ$, and altitude AD = $5 \text{ cm}$ (D on BC). He draws a line, marks D, constructs a perpendicular at D, and marks A such that AD = $5 \text{ cm}$. To find points B and C on the line through D, he needs to use angles $\angle B$ and $\angle C$. What angles should he construct at A (on the perpendicular)?

(A) Construct $\angle ADB = 90^\circ$ and $\angle ADC = 90^\circ$.

(B) Construct $\angle DAB = 90^\circ - 60^\circ = 30^\circ$ and $\angle DAC = 90^\circ - 45^\circ = 45^\circ$. The arms intersect the line at B and C.

(C) Construct angles equal to $60^\circ$ and $45^\circ$ at A.

(D) Draw lines parallel to the perpendicular at angles $60^\circ$ and $45^\circ$ from B and C.

Question 1. Pooja needs to construct a triangle similar to $\triangle ABC$ with a scale factor of $2/3$. She draws the base BC and a ray BX from B making an acute angle. How many equal parts should she mark on ray BX using her compass?

(A) 2

(B) 3

(C) 5

(D) 6

Question 2. For the scale factor $2/3$, Pooja marks 3 equal parts $B_1, B_2, B_3$ on ray BX. She connects $B_3$ to C. To find the corresponding point C' on BC for the similar triangle, what should she do next?

(A) Draw a line through $B_2$ parallel to $B_3C$, intersecting BC at C'.

(B) Draw a line through $B_3$ parallel to AC, intersecting BC at C'.

(C) Draw a line through $B_2$ perpendicular to BC.

(D) Measure BC and mark $2/3$ of its length as BC'.

Question 3. Now Pooja needs to construct a similar triangle with a scale factor of $3/2$. She again draws BC and ray BX. How many equal parts should she mark on ray BX, and which point should she connect to C first?

(A) Mark 2 parts, connect $B_2$ to C.

(B) Mark 3 parts, connect $B_3$ to C.

(C) Mark 3 parts, connect $B_2$ to C.

(D) Mark 2 parts, connect $B_3$ to C.

Question 4. For the scale factor $3/2$, Pooja marks 3 equal parts and connects $B_2$ to C. To find C' for the larger triangle, she draws a line through $B_3$ parallel to $B_2C$. Where will C' be located?

(A) On the segment BC.

(B) On the extension of BC beyond C.

(C) On the extension of BC beyond B.

(D) Coinciding with C.

Question 5. Pooja needs to justify why her constructed triangle is similar to the original $\triangle ABC$ with the desired scale factor. Which geometric theorem is fundamental to this justification?

(A) Pythagorean Theorem.

(B) Angle Sum Property.

(C) Basic Proportionality Theorem (Thales Theorem).

(D) Triangle Congruence Criteria.

Question 6. After finding C' on BC (or its extension), Pooja needs to find the corresponding vertex A'. She draws a line through C' parallel to AC. Which angle property guarantees that the new triangle $\triangle A'BC'$ has the same angles as $\triangle ABC$ at A and C (i.e., $\angle BA'C' = \angle BAC$ and $\angle BC'A' = \angle BCA$)?

(A) Vertically Opposite Angles are equal.

(B) Alternate Interior Angles are equal.

(C) Corresponding Angles are equal.

(D) Angles on a straight line sum to $180^\circ$.

Question 7. Pooja constructed a triangle with a scale factor of 2. If the perimeter of the original triangle was $15 \text{ cm}$, what is the perimeter of the constructed similar triangle?

(A) $15 \text{ cm}$

(B) $30 \text{ cm}$

(C) $60 \text{ cm}$

(D) $7.5 \text{ cm}$

Question 1. Rajesh needs to construct a general quadrilateral ABCD given sides AB=$5 \text{ cm}$, BC=$6 \text{ cm}$, CD=$7 \text{ cm}$, DA=$8 \text{ cm}$, and diagonal AC=$9 \text{ cm}$. He first draws the diagonal AC. What criterion should he use to construct $\triangle ABC$ and $\triangle ADC$ on opposite sides of AC?

(A) SAS for both triangles.

(B) ASA for both triangles.

(C) SSS for both triangles.

(D) RHS for both triangles.

Question 2. Rajesh needs to construct a parallelogram given adjacent sides $5 \text{ cm}$ and $7 \text{ cm}$, and the included angle $60^\circ$. Which triangle construction criterion can he use as the first step to construct one part of the parallelogram divided by an imaginary diagonal?

(A) SSS

(B) SAS

(C) ASA

(D) AAS

Question 3. Rajesh needs to construct a rectangle with length $8 \text{ cm}$ and width $5 \text{ cm}$. He draws a segment of length $8 \text{ cm}$. At one endpoint, he constructs a $90^\circ$ angle. What is the next step to complete the construction of the rectangle using compass and ruler?

(A) Draw an arc of radius $5 \text{ cm}$ from the other endpoint of the $8 \text{ cm}$ segment.

(B) Mark a point on the arm of the $90^\circ$ angle at a distance of $5 \text{ cm}$ from the vertex.

(C) Construct a $90^\circ$ angle at the other endpoint of the $8 \text{ cm}$ segment.

(D) Both (B) and (C) are necessary steps in the construction.

Question 4. Rajesh needs to construct a rhombus given its two diagonals are $6 \text{ cm}$ and $8 \text{ cm}$. Which property of a rhombus's diagonals is most useful for this construction?

(A) Diagonals are equal.

(B) Diagonals are perpendicular bisectors of each other.

(C) Diagonals bisect the vertex angles.

(D) The sum of the squares of the diagonals equals four times the square of the side.

Question 5. Rajesh is asked to construct a square with side length $6 \text{ cm}$. Which of the following is a valid method using compass and ruler?

(A) Draw a segment of $6 \text{ cm}$, construct $90^\circ$ at both ends, mark $6 \text{ cm}$ on perpendiculars, join points.

(B) Draw a segment of $6 \text{ cm}$, construct a $60^\circ$ angle at one end, mark $6 \text{ cm}$ on the arm, draw arcs of $6 \text{ cm}$ from the other two vertices.

(C) Draw a segment of $6\sqrt{2} \text{ cm}$ (diagonal), construct its perpendicular bisector, mark $3 \text{ cm}$ on either side of the midpoint along the bisector.

(D) Draw a segment of $6 \text{ cm}$, construct a $90^\circ$ angle at one end, draw an arc of radius $6\sqrt{2}$ from the other end.

Question 6. Rajesh is given the lengths of the four sides of a quadrilateral. Can he always construct a unique quadrilateral with just this information?

(A) Yes, always.

(B) No, the shape is not unique; it can vary depending on the angles (e.g., different parallelograms or kites with the same side lengths).

(C) Yes, but only if it's a special type like a square or rhombus.

(D) No, unless at least one angle or a diagonal length is also given.

Question 1. Suman is given a circle with center O and a point P on the circle. She needs to construct the tangent at P. She draws the radius OP. What is the correct next step using compass and ruler?

(A) Draw a line parallel to OP through P.

(B) Construct a line perpendicular to the radius OP at the point P.

(C) Draw a line segment from O to any other point on the circle.

(D) Draw a circle centered at P with radius OP.

Question 2. Suman needs to construct tangents from a point Q outside the circle with center O. She joins O to Q. She finds the midpoint M of OQ by constructing its perpendicular bisector. What is the crucial next step in the standard construction method?

(A) Draw a circle with center O and radius OQ.

(B) Draw a circle with center Q and radius OQ.

(C) Draw a circle with center M and radius OM (or MQ).

(D) Draw a line through M perpendicular to OQ.

Question 3. In Suman's construction of tangents from an external point Q, the auxiliary circle (with diameter OQ) intersects the original circle at points R and S. She joins Q to R and Q to S. Which geometric property justifies that QR and QS are the tangents?

(A) The lengths of tangents from an external point are equal.

(B) The angle subtended by a diameter in a semicircle is $90^\circ$, making radii OR and OS perpendicular to QR and QS.

(C) Triangle ORQ and OSQ are congruent.

(D) The line OQ bisects the angle $\angle RQS$.

Question 4. Suman needs to construct a pair of tangents from an external point such that the angle between them is $60^\circ$. Let O be the center and the external point be P. Let the points of tangency be A and B. The angle between the tangents is $\angle APB = 60^\circ$. What is the measure of the angle $\angle AOB$ between the radii to the points of contact?

(A) $60^\circ$

(B) $90^\circ$

(C) $120^\circ$

(D) $180^\circ$

Question 5. Suman constructs the tangent at point P on the circle. Her teacher asks her to justify *why* the line she drew is a tangent. What is the definition of a tangent she should use in her justification?

(A) A line that passes through the center of the circle.

(B) A line that intersects the circle at exactly one point.

(C) A line that is perpendicular to the radius at any point.

(D) A line segment whose endpoints are on the circle.

Question 6. Suman is given a circle without its center marked. She needs to construct tangents from an external point P. What must she do first?

(A) Draw a line through P.

(B) Estimate the center of the circle.

(C) Find the center of the circle using a compass and ruler construction.

(D) Draw an arc from P intersecting the circle.

Question 1. The teacher asks students to justify their constructions. For example, justifying the angle bisector construction. What is the primary purpose of this justification?

(A) To check if the drawing is neat and accurate.

(B) To prove that the constructed figure satisfies the properties it is supposed to have, based on geometric principles.

(C) To describe the step-by-step process of the construction.

(D) To show that the construction can be done quickly.

Question 2. When justifying the perpendicular bisector construction, students often refer to the property that any point on the bisector is equidistant from the segment's endpoints. How is this property typically proven in Euclidean geometry?

(A) By measuring distances with a ruler.

(B) By assuming it is true (as an axiom).

(C) By using triangle congruence criteria (like SSS or SAS) to show that triangles formed are congruent, implying equal corresponding sides.

(D) Using the Pythagorean theorem.

Question 3. A student justifies the construction of parallel lines using the statement: "If corresponding angles are equal, then the lines are parallel." What type of geometric reasoning is this statement?

(A) A definition.

(B) An axiom or postulate.

(C) A theorem.

(D) The converse of a theorem/postulate.

Question 4. To justify the construction of dividing a line segment in a given ratio, students use the Basic Proportionality Theorem (BPT). What is the fundamental idea of BPT that makes it applicable here?

(A) Angles in similar triangles are equal.

(B) Parallel lines cut transversals proportionally.

(C) Congruent triangles have equal corresponding parts.

(D) The square of the hypotenuse equals the sum of the squares of the other two sides.

Question 5. A student uses a protractor to check if the angle they constructed is $60^\circ$. They find it is $59.8^\circ$. Does this measurement constitute a formal justification of their $60^\circ$ construction method?

(A) Yes, if the measurement is close enough.

(B) Yes, measurement is a valid form of proof in geometry.

(C) No, measurement is a way to verify the accuracy of the drawing, but not a formal proof based on logical deduction from geometric principles.

(D) No, because the measurement is not exactly $60^\circ$.

Question 6. The process of geometric justification helps students develop crucial mathematical skills. Which of the following skills are enhanced through justification?

(A) Rote memorization of construction steps.

(B) Artistic drawing abilities.

(C) Logical reasoning and deductive thinking.

(D) Estimation of lengths and angles.