Chapter 6 The Triangle and its Properties (Concepts)

Triangle and Types of Triangles

In geometry, one of the most fundamental and widely studied polygons is the Triangle. Triangles are simple yet incredibly versatile shapes, found everywhere in the world around us, from the stability provided by triangular structures in buildings and bridges to patterns in nature and design. Understanding triangles and their properties is essential for further study in geometry and other areas of mathematics and science.

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What is a Triangle?

A Triangle is a closed two-dimensional plane figure formed by three non-parallel and non-collinear line segments that connect three distinct points. It is the polygon with the fewest number of sides.

Every triangle has three main components:

• Sides: The three line segments that form the boundary of the triangle.
• Vertices: The three points where the sides meet or intersect. These are the 'corners' of the triangle.
• Angles: The three angles formed inside the triangle at each vertex by the intersection of the two sides meeting at that vertex.

In the figure above, the triangle has vertices A, B, and C. Its sides are the line segments AB, BC, and CA. The angles are $\angle A$ (or $\angle BAC$), $\angle B$ (or $\angle ABC$), and $\angle C$ (or $\angle BCA$). A triangle is denoted by the symbol $\triangle$. So, the triangle with vertices A, B, and C is denoted as $\triangle ABC$.

Elements of a Triangle:

A triangle has a total of six elements (or parts):

• 3 sides
• 3 angles

Interior and Exterior of a Triangle:

• The region inside the triangle, enclosed by its three sides, is called the interior of the triangle.
• The region outside the triangle is called the exterior of the triangle.
• The three line segments that form the triangle itself make up its boundary.

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Types of Triangles

Triangles can be classified into different types based on their properties. The two most common ways to classify triangles are based on the lengths of their sides and the measures of their angles.

1. Classification Based on Sides

Based on the relative lengths of their three sides, triangles are classified into three main types:

(a) Equilateral Triangle:

• An equilateral triangle is a triangle in which all three sides are equal in length.
• As a direct consequence of having equal sides, all three interior angles of an equilateral triangle are also equal. Each angle in an equilateral triangle measures exactly $60^\circ$.

If in $\triangle ABC$, $AB = BC = CA$.

Then $m\angle A = m\angle B = m\angle C = 60^\circ$.

(b) Isosceles Triangle:

• An isosceles triangle is a triangle in which at least two sides are equal in length.
• The angles opposite the two equal sides are also equal. These equal angles are often called the base angles.

If in $\triangle PQR$, side $PQ$ is equal in length to side $PR$ ($PQ = PR$).

Then the angle opposite PQ ($\angle R$) is equal to the angle opposite PR ($\angle Q$), i.e., $m\angle R = m\angle Q$.

Note: An equilateral triangle is also a special case of an isosceles triangle, as it satisfies the condition of having at least two equal sides (in fact, it has three).

(c) Scalene Triangle:

• A scalene triangle is a triangle in which all three sides have different lengths.
• Consequently, all three interior angles of a scalene triangle are also different in measure. The largest angle is opposite the longest side, and the smallest angle is opposite the shortest side.

2. Classification Based on Angles

Based on the measures of their interior angles, triangles are classified into three types:

(a) Acute-angled Triangle (or Acute Triangle):

• An acute-angled triangle is a triangle in which all three interior angles are acute angles. An acute angle is an angle that measures less than $90^\circ$.

(b) Right-angled Triangle (or Right Triangle):

• A right-angled triangle is a triangle in which exactly one of the interior angles is a right angle. A right angle measures exactly $90^\circ$.
• The side opposite the right angle is always the longest side of a right-angled triangle and is called the hypotenuse.
• The other two sides, which form the right angle, are called the legs or perpendicular sides.
• The two angles other than the right angle are always acute angles, and they are complementary (their sum is $90^\circ$).

(c) Obtuse-angled Triangle (or Obtuse Triangle):

• An obtuse-angled triangle is a triangle in which exactly one of the interior angles is an obtuse angle. An obtuse angle is an angle that measures greater than $90^\circ$ but less than $180^\circ$.
• A triangle can have at most one obtuse angle. If one angle is obtuse, the sum of the other two angles must be less than $90^\circ$, meaning the other two angles must be acute.

It is common to classify a triangle by both its side lengths and angle measures. For example, a triangle could be a "scalene right-angled triangle", an "isosceles acute-angled triangle", or even an "isosceles right-angled triangle". An equilateral triangle is always acute-angled.

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Special Line Segments in a Triangle

In addition to sides and angles, there are special line segments associated with triangles that have important properties.

Median of a Triangle

A Median of a triangle is a line segment that connects a vertex to the mid-point of the opposite side. Since a triangle has three vertices, it has three medians.

• In $\triangle ABC$ below, the point D is the mid-point of the side BC. The line segment AD is the median from vertex A to side BC.
• A triangle has three medians.
• The three medians of a triangle always intersect at a single point inside the triangle, called the centroid. The centroid is the 'center of gravity' of the triangle.

Altitude of a Triangle

An Altitude of a triangle is the perpendicular line segment from a vertex to the opposite side (or to the line containing the opposite side). The altitude represents the height of the triangle with respect to the side it is drawn to, which is called the base.

• In $\triangle PQR$ below, the line segment PM is perpendicular to the side QR. So, PM is the altitude from vertex P to the base QR.
• A triangle has three altitudes, one from each vertex.
• The position of the altitude can be inside, outside, or on the triangle itself, depending on whether the triangle is acute, obtuse, or right-angled.
• The three altitudes of a triangle intersect at a single point called the orthocenter.

(a) Altitude in an Acute Triangle: All three altitudes lie inside the triangle.

(b) Altitude in a Right-angled Triangle: Two of the altitudes are the legs (perpendicular sides) of the triangle itself. The third altitude, from the right-angle vertex to the hypotenuse, lies inside the triangle. The orthocenter is the vertex where the right angle is located.

(c) Altitude in an Obtuse-angled Triangle: The altitude from the obtuse vertex lies inside the triangle. The two altitudes from the acute vertices lie outside the triangle, as the opposite sides must be extended to draw the perpendiculars.

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Properties of a Triangle

Triangles are not just shapes formed by three sides; they possess many interesting and useful properties related to their angles, sides, and other special lines drawn within them. Understanding these properties is fundamental to solving various problems in geometry.

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1. Angle Sum Property of a Triangle

This is one of the most important properties of any triangle.

Statement: The sum of the measures of the three interior angles of any triangle is always equal to $180^\circ$. This property holds true for all types of triangles (acute, right, obtuse, equilateral, isosceles, scalene).

If in $\triangle ABC$, the measures of the interior angles at vertices A, B, and C are $m\angle A, m\angle B,$ and $m\angle C$ respectively, then:

Proof of the Angle Sum Property:

We can prove this property using the concepts of parallel lines and transversals that we learned in the previous chapter.

Given: A triangle $\triangle ABC$.

To Prove: $m\angle BAC + m\angle ABC + m\angle BCA = 180^\circ$.

Let's label the angles inside $\triangle ABC$ as $m\angle BAC = \angle 1, m\angle ABC = \angle 2,$ and $m\angle BCA = \angle 3$. We need to prove $m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$.

Construction: Through the vertex A, draw a straight line XAY that is parallel to the side BC. This line XAY passes through A.

Proof:

Since XAY is a straight line, the sum of the angles formed on this line at point A is $180^\circ$.

Looking at the figure, the angles on the straight line XAY at point A are $\angle XAB, \angle BAC,$ and $\angle YAC$.

Now, consider the parallel lines XAY and BC, and the transversal AB. The angles $\angle XAB$ and $\angle ABC$ are alternate interior angles.

Similarly, consider the parallel lines XAY and BC, and the transversal AC. The angles $\angle YAC$ and $\angle BCA$ are alternate interior angles.

Now, substitute the equalities from (ii) and (iii) into equation (i):

Replace $m\angle XAB$ with $m\angle ABC$ and $m\angle YAC$ with $m\angle BCA$ in equation (i).

Rearranging the terms to match the usual order of angles:

This proves that the sum of the measures of the three interior angles of any triangle is always $180^\circ$. (Hence Proved)

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2. Exterior Angle Property of a Triangle

When we extend one side of a triangle, an angle is formed outside the triangle. This angle is called an exterior angle. There is a special relationship between an exterior angle and the interior angles of the triangle.

An Exterior Angle of a triangle is formed when one side of the triangle is extended outwards. The exterior angle is adjacent to one of the interior angles of the triangle, and together they form a linear pair ($180^\circ$).

In $\triangle ABC$, if the side BC is extended to a point D, then $\angle ACD$ is an exterior angle of $\triangle ABC$ at vertex C. The angles $\angle BAC$ (or $\angle A$) and $\angle ABC$ (or $\angle B$) are the two interior angles that are not adjacent to $\angle ACD$. These are called the interior opposite angles (or remote interior angles) with respect to the exterior angle $\angle ACD$. $\angle ACB$ (or $\angle 3$ in the previous proof) is the interior adjacent angle to $\angle ACD$.

Statement (Exterior Angle Property): The measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.

In the figure above, the exterior angle $\angle ACD$ is equal to the sum of the interior opposite angles $\angle A$ and $\angle B$.

Proof of the Exterior Angle Property:

We can prove this property using the Angle Sum Property of a Triangle and the Linear Pair property.

Given: $\triangle ABC$ with side BC extended to D, forming exterior angle $\angle ACD$. Let $m\angle BAC = \angle 1, m\angle ABC = \angle 2, m\angle ACB = \angle 3,$ and $m\angle ACD = \angle 4$.

To Prove: $m\angle 4 = m\angle 1 + m\angle 2$.

Proof:

In $\triangle ABC$, by the Angle Sum Property:

Now, look at the straight line BCD. The angles $\angle ACB$ ($\angle 3$) and $\angle ACD$ ($\angle 4$) are adjacent angles on this straight line, forming a linear pair.

From equations (i) and (ii), both $(m\angle 1 + m\angle 2 + m\angle 3)$ and $(m\angle 3 + m\angle 4)$ are equal to $180^\circ$. Therefore, they are equal to each other.

Subtract $m\angle 3$ from both sides of this equation:

Or, writing in terms of vertex labels: $m\angle BAC + m\angle ABC = m\angle ACD$.

This proves that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles. (Hence Proved)

Corollary: An exterior angle of a triangle is always greater than either of its interior opposite angles. This is because the exterior angle is the sum of two positive angle measures. So, $m\angle ACD > m\angle A$ and $m\angle ACD > m\angle B$.

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3. Medians of a Triangle

A Median of a triangle is a line segment drawn from a vertex of the triangle to the midpoint of the side opposite that vertex.

In $\triangle ABC$, if D is the midpoint of side BC, then the line segment AD is the median from vertex A to side BC. Similarly, the median from vertex B connects B to the midpoint of AC, and the median from vertex C connects C to the midpoint of AB.

• Every triangle has exactly three medians, one originating from each vertex.
• The three medians of a triangle are concurrent, meaning they all intersect at a single point.
• This point of concurrency of the medians is called the Centroid of the triangle. The centroid is the triangle's center of mass or balance point.

In the figure, AD, BE, and CF are the medians of $\triangle ABC$, and they intersect at point G, which is the centroid.

(The property that the centroid divides each median in the ratio 2:1 is typically covered in higher classes.)

4. Altitudes of a Triangle

An Altitude of a triangle is a perpendicular line segment drawn from a vertex of the triangle to the side opposite that vertex, or to the extension of the opposite side if necessary. The altitude represents the 'height' of the triangle relative to the side to which it is drawn (that side is considered the 'base').

In $\triangle ABC$, AD is the altitude from vertex A to side BC if AD is perpendicular to BC ($AD \perp BC$). Similarly, BE is the altitude from B to AC if $BE \perp AC$, and CF is the altitude from C to AB if $CF \perp AB$.

• Every triangle has exactly three altitudes, one from each vertex.
• The three altitudes of a triangle are also concurrent, meaning they all intersect at a single point.
• This point of concurrency of the altitudes is called the Orthocenter of the triangle.
• The position of the orthocenter depends on the type of triangle:
  • For an acute-angled triangle, the orthocenter lies inside the triangle. (As shown in the figure above).
  • For a right-angled triangle, the orthocenter is located exactly at the vertex where the right angle is formed. (The two legs of the right triangle are themselves altitudes).
  • For an obtuse-angled triangle, the orthocenter lies outside the triangle. To draw the altitude from a vertex in an obtuse triangle, you might need to extend the opposite side.

In the figure above, for obtuse $\triangle PQR$, the altitude from R is drawn perpendicular to the extension of side PQ at point S. The orthocenter would be outside the triangle.

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Angle Property of Special Triangles

We have already discussed the general properties of all triangles, such as the Angle Sum Property and the Exterior Angle Property. Now, let's look at some special types of triangles that have specific properties related to their angles due to their unique side lengths. These special triangles are the isosceles triangle and the equilateral triangle.

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1. Angles of an Isosceles Triangle

An Isosceles Triangle is a triangle that has at least two sides of equal length. Because of this equality in side lengths, there is a corresponding equality in the measures of certain angles.

Property: In an isosceles triangle, the angles that are opposite to the two equal sides are equal in measure. These equal angles are often called the base angles of the isosceles triangle.

If in $\triangle ABC$, side AB has the same length as side AC ($AB = AC$), then the angle opposite to side AB, which is $\angle C$, is equal in measure to the angle opposite to side AC, which is $\angle B$.

The vertex where the two equal sides meet (vertex A in the figure) is called the vertex angle. The side opposite the vertex angle (side BC) is called the base. The angles at the ends of the base ($\angle B$ and $\angle C$) are the base angles.

Converse Property: The reverse of this property is also true. In a triangle, if two angles are equal in measure, then the sides opposite to these equal angles are also equal in length. If this property holds, the triangle is an isosceles triangle.

Proof that angles opposite equal sides are equal:

Let's attempt a proof using a standard construction, typically involving congruence of triangles, which you might learn more about later. However, the reasoning can be understood at a basic level.

Given: $\triangle ABC$ such that side $AB$ is equal to side $AC$ ($AB = AC$).

To Prove: $m\angle B = m\angle C$.

Construction: Draw the bisector of $\angle A$, let's call it AD, such that point D lies on side BC. This line segment AD divides $\angle A$ into two equal angles, $\angle BAD$ and $\angle CAD$.

Proof:

Consider the two triangles formed by the construction: $\triangle ABD$ and $\triangle ACD$.

We can compare these two triangles:

• Side AB is equal to side AC ($AB = AC$).
• Angle $\angle BAD$ is equal to angle $\angle CAD$ ($m\angle BAD = m\angle CAD$) because AD is the angle bisector of $\angle A$.
• Side AD is common to both triangles ($AD = AD$).

Based on these three equal parts (Side-Angle-Side), we can say that $\triangle ABD$ is congruent to $\triangle ACD$. Congruent triangles are identical in shape and size; one can be perfectly superimposed on the other.

Since the triangles are congruent, their corresponding parts are equal. The angle $\angle B$ in $\triangle ABD$ corresponds to the angle $\angle C$ in $\triangle ACD$.

This proves that the angles opposite the equal sides in an isosceles triangle are equal. (Hence Proved)

(Note: An alternative proof for $m\angle B = m\angle C$ when $AB=AC$ can be done by constructing the perpendicular bisector of BC, which for an isosceles triangle from A will pass through A and also bisect angle A.)

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2. Angles of an Equilateral Triangle

An Equilateral Triangle is a triangle in which all three sides are equal in length. It is a special case of an isosceles triangle because it has at least two equal sides (in fact, it has three).

Property: In an equilateral triangle, all three interior angles are equal in measure. Each angle in an equilateral triangle measures exactly $60^\circ$.

If in $\triangle PQR$, $PQ = QR = RP$, then $m\angle P = m\angle Q = m\angle R = 60^\circ$.

Derivation of the Angle Measure:

Let $\triangle PQR$ be an equilateral triangle. By definition, $PQ = QR = RP$.

Since $PQ = QR$, by the property of isosceles triangles (angles opposite equal sides are equal), the angle opposite PQ ($\angle R$) is equal to the angle opposite QR ($\angle P$).

Since $QR = RP$, by the property of isosceles triangles, the angle opposite QR ($\angle P$) is equal to the angle opposite RP ($\angle Q$).

Combining these equalities, we get $m\angle P = m\angle Q = m\angle R$. Let the measure of each of these equal angles be $x$.

By the Angle Sum Property of a triangle, the sum of the three interior angles is $180^\circ$.

Substitute $x$ for each angle measure:

Divide both sides by 3 to solve for $x$:

Thus, each angle of an equilateral triangle measures $60^\circ$.

Converse Property: The converse is also true. If all three angles of a triangle are equal in measure (each measuring $60^\circ$), then all three sides are equal in length, and the triangle is equilateral.

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Sum of Lengths of Two Sides of a Triangle

In our exploration of triangles, we have looked at their classification and some angle properties. Now, let's examine a crucial property that relates the lengths of the sides of any triangle. This property, known as the Triangle Inequality, tells us whether it is even possible to form a triangle given three specific side lengths. It also establishes a fundamental relationship between the sides within any triangle.

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Triangle Inequality Property

The Triangle Inequality Property is a fundamental theorem in geometry.

Statement: The sum of the lengths of any two sides of a triangle is always strictly greater than the length of the third side.

Consider a triangle $\triangle ABC$ with vertices A, B, and C. Let the length of the side opposite vertex A (side BC) be denoted by $a$. Let the length of the side opposite vertex B (side AC) be denoted by $b$. Let the length of the side opposite vertex C (side AB) be denoted by $c$.

According to the Triangle Inequality Property, the following three inequalities must always be true for any triangle:

1. The sum of side $a$ and side $b$ must be greater than side $c$: $a + b > c$.
2. The sum of side $b$ and side $c$ must be greater than side $a$: $b + c > a$.
3. The sum of side $c$ and side $a$ must be greater than side $b$: $c + a > b$.

If, for any given set of three lengths, even one of these three conditions is not satisfied, then those three lengths cannot form a triangle. The three line segments would not be able to connect to form a closed three-sided figure.

Intuitive Explanation:

The Triangle Inequality makes intuitive sense. Think about traveling between two points. The shortest distance between any two points is a straight line. If you have three points P, Q, and R that form a triangle, going directly from P to R is a straight line segment (side PR). If you go from P to R by first going to Q and then to R (forming sides PQ and QR), the total distance traveled $(PQ + QR)$ must be longer than the direct straight path (PR), unless P, Q, and R lie on the same straight line. If P, Q, and R are collinear, then $PQ + QR = PR$, but they would just form a line segment, not a triangle.

The Triangle Inequality captures this geometric fact: the sum of the lengths of two sides must "reach past" the third side for the vertices to connect and form a triangle.

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Checking if a Triangle Can Be Formed

To determine quickly if three given lengths can form a triangle, you don't necessarily need to check all three inequalities $a+b>c$, $b+c>a$, and $c+a>b$. It is sufficient to check only one condition:

Let the three given lengths be arranged in non-decreasing order, say $l_1 \le l_2 \le l_3$ (where $l_3$ is the longest side, or equal to the longest side if there are equal sides). A triangle can be formed with these three lengths if and only if the sum of the lengths of the two shorter sides is strictly greater than the length of the longest side.

Rule: Given three lengths $l_1, l_2, l_3$, arranged such that $l_1 \le l_2 \le l_3$. A triangle can be formed if and only if $l_1 + l_2 > l_3$. If this single condition holds, the other two conditions of the Triangle Inequality Property will automatically hold ($l_2+l_3 > l_1$ and $l_1+l_3 > l_2$ will be true because $l_3$ is the largest, so adding anything positive to it will result in a sum larger than $l_1$ or $l_2$).

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Relationship between Sides and Angles

While not directly part of the Triangle Inequality property itself, there is an important relationship between the lengths of the sides of a triangle and the measures of the angles opposite to those sides:

• In any triangle, the angle opposite the longest side is the largest angle in the triangle.
• In any triangle, the angle opposite the shortest side is the smallest angle in the triangle.
• If two sides of a triangle are equal, then the angles opposite those sides are equal (as in an isosceles triangle). (This is the isosceles triangle property).
• Conversely, if two angles of a triangle are equal, then the sides opposite those angles are equal.

This relationship is always true and complements the Triangle Inequality Property in describing how sides and angles determine each other in a triangle.

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Pythagoras Property of a Right Angled Triangle

Among the different types of triangles, the Right-Angled Triangle holds special significance in geometry. It is defined by having one angle that measures exactly $90^\circ$. The sides of a right-angled triangle have a unique and extremely important relationship among their lengths, which is described by the Pythagoras Property (also widely known as the Pythagorean Theorem). This property is named after the ancient Greek mathematician Pythagoras, although the relationship was likely known in earlier civilisations like the Babylonians and Egyptians.

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Right-Angled Triangle Recap

A Right-Angled Triangle is a triangle where one of its three interior angles is a right angle ($90^\circ$). The side opposite the right angle is the longest side of the triangle, and it has a special name.

In a right-angled triangle:

• The side directly opposite the $90^\circ$ angle is called the Hypotenuse. It is always the longest side of the right triangle.
• The other two sides, which form the right angle, are called the Legs of the triangle. Sometimes, they are referred to as the base and the perpendicular or height, depending on how the triangle is oriented or used in a problem.

In the figure above, if $\angle C = 90^\circ$, then side AB is the hypotenuse, and sides AC and BC are the legs.

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Statement of the Pythagoras Property (Pythagorean Theorem)

The Pythagoras property describes the relationship between the lengths of the hypotenuse and the two legs of any right-angled triangle.

Statement: In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (the legs).

Let the length of the hypotenuse be $c$, and the lengths of the two legs be $a$ and $b$. The property can be expressed by the formula:

In terms of the sides of the triangle in the figure where $\angle C = 90^\circ$, the hypotenuse is AB, and the legs are AC and BC. So, the property is:

This can be remembered as: $$(\text{length of Hypotenuse})^2 = (\text{length of Leg } 1)^2 + (\text{length of Leg } 2)^2$$

Visual Representation:

The Pythagoras property can be beautifully illustrated visually. If you draw a square on each side of a right-angled triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the two legs.

If the legs have lengths $a$ and $b$, and the hypotenuse has length $c$, the areas of the squares on the sides are $a^2$, $b^2$, and $c^2$. The property $a^2 + b^2 = c^2$ means Area of square on leg 1 + Area of square on leg 2 = Area of square on hypotenuse.

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Converse of the Pythagoras Property

The converse of the Pythagoras property is also a very important statement. It allows us to determine if a triangle is right-angled when we only know the lengths of its three sides.

Statement (Converse): If in any triangle, the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides, then the angle opposite the longest side is a right angle ($90^\circ$), and therefore, the triangle is a right-angled triangle.

If in $\triangle ABC$, with side lengths $a, b, c$, it is found that $c^2 = a^2 + b^2$, where $c$ is the longest side, then the angle opposite side $c$ (which is $\angle C$) must be $90^\circ$.

This property is used to check if a given triangle is a right triangle by examining its side lengths.

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Applications of Pythagoras Property

The Pythagoras property is fundamental in many areas:

• Finding Unknown Side Lengths: If you know the lengths of two sides of a right-angled triangle, you can use the Pythagoras property to calculate the length of the third side.
  • If you know the lengths of the two legs ($a$ and $b$), you can find the hypotenuse ($c$) using: $c = \sqrt{a^2 + b^2}$.
  • If you know the length of the hypotenuse ($c$) and one leg ($a$), you can find the other leg ($b$) using: $b = \sqrt{c^2 - a^2}$. Similarly, if you know $c$ and leg $b$, $a = \sqrt{c^2 - b^2}$. (Remember, the hypotenuse is always the longest side, so $c^2$ will be larger than $a^2$ or $b^2$).
• Checking for Right Triangles: Use the converse of the property to determine if a triangle is right-angled given its three side lengths.
• Solving Problems: It is used extensively in physics (e.g., vector addition), engineering, architecture, navigation, computer graphics, and many other fields where distances and right angles are involved.

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