Right Angled Triangle: Definition, Properties and Solved Examples for Class 7

A right-angled triangle is a special type of triangle in which one of the angles is exactly 90 degrees. It is also commonly called a right triangle. The term ‘right-angled triangle’ is commonly used in British English, while ‘right triangle’ is more commonly used in American English. Both terms refer to the same geometric shape and are widely accepted in mathematics. This shape forms the foundation for many concepts in geometry and trigonometry. In this topic, you will learn the basic structure of a right-angled triangle, its different parts like hypotenuse, base, and height, and how it is used in mathematical calculations with clear explanations and examples for better understanding.

Table of Contents

• What is a Right Angled Triangle?
• The Three Sides of a Right Angled Triangle
• Properties of a Right Triangle
• Pythagoras's Theorem
• Area and Perimeter of a Right Triangle
• Solved Examples on Right Angled Triangle
• Practice Questions on Right Angled Triangle

What is a Right Angled Triangle?

A right-angled triangle is a triangle in which one of the three angles is exactly 90°. This 90° angle is called the right angle, and the small square symbol (□) is used to mark it in diagrams. The remaining two angles are both acute (less than 90°) and they always add up to exactly 90°.

You've seen them everywhere, like in the corner of your notebook, in the ramp leading up to a building, in a slice of pizza, or in the way a ladder leans against a wall.

The Three Sides of a Right Angled Triangle

Each side of a right-angled triangle has a specific name and role, and these names also form the basis of trigonometry.

 

Properties of a Right Triangle

The right triangle has several unique properties that distinguish it from all other triangles.

• One angle is exactly 90°: This is the defining property. If even one angle is 90°, the triangle is a right triangle.

• The other two angles are acute and complementary: They are both less than 90° and together they sum to exactly 90°. If one is 30°, the other must be 60°.

• The hypotenuse is the longest side. : It is always longer than either of the two legs. This follows directly from the fact that the largest side is always opposite the largest angle and 90° is the largest angle.

• Pythagoras' theorem holds: For a right triangle with legs a and b and hypotenuse c: c² = a² + b²

• The sum of all three angles is 180°: like all triangles. Since one angle is 90°, the remaining two together make up the other 90°.

• The circumcircle's centre lies on the hypotenuse: In a right triangle, the hypotenuse is the diameter of the circumscribed circle (the circle passing through all three vertices). 

• A right triangle cannot be an obtuse triangle: Since one angle is already 90°, no other angle can exceed 90° (the sum would exceed 180°).

• The median to the hypotenuse equals half the hypotenuse: In a right triangle, the median drawn from the right-angle vertex to the hypotenuse is exactly half the length of the hypotenuse.

Pythagoras's Theorem

The Pythagorean theorem states that in a right-angled triangle

Hypotenuse² = Base² + Perpendicular²

Written as: c² = a² + b²

where c = hypotenuse, a = one leg (base), b = other leg (perpendicular)

Equivalently: c = √(a² + b²)

In plain English: the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides.

This result can be understood using a simple geometric arrangement of four identical triangles inside a square.

In a right-angled triangle ∆ABC, B = 90°

Let AB = a, BC = b and AC = c.

Arrange four such triangles inside a square in two ways as shown in the diagram below.

• In the 1st arrangement, we can observe that there is a square of side c, which is uncovered.

• In the 2nd arrangement, we can observe that there are two squares of sides a and b, which are uncovered.

As both arrangements are done using four equal triangles in the same square, the areas of the uncovered parts, in both cases, are equal.

uncovered area in the 1st arrangement = sum of uncovered areas in the 2nd arrangement

area of uncovered square in the 1st arrangement = Total area of two uncovered squares in the 2nd arrangement

c² = a² + b² (area of a square = side²)

This is the relation given by the Pythagorean theorem.

In a right-angled triangle, the square of the hypotenuse equals to the sum of the squares of the other two sides (or legs).

Pythagorean Triples

A Pythagorean triplet is a set of three positive integers (a, b, and c) that satisfy the Pythagorean theorem. If a, b and c are any three natural numbers and they satisfy the relation of the Pythagorean theorem, then a, b and c form a Pythagorean triplet.

Example: (3, 4, 5), (6, 8, 10), (5, 12, 13), etc.

Area and Perimeter of a Right Triangle

Area of a Right-Angled Triangle: 

Area = ½ × Base × Height

= ½ × b × h

where b = base (one leg) and h = height (the perpendicular side)

The unit is always square units (cm², m², etc.)

Perimeter of a Right Triangle:

P = a + b + c

where a and b are the two legs (base and perpendicular) and c is the hypotenuse

Since c = √(a² + b²), you can also write P = a + b + √(a² + b²)

Solved Examples on Right Angled Triangle

Example 1: The two legs of a right triangle are 6 cm and 8 cm. Find the length of the hypotenuse.

Solution: Given a = 6 cm, b = 8 cm, c = hypotenuse = ?

Apply the Pythagorean theorem: c² = a² + b² = 6² + 8² = 36 + 64 = 100

c = √100 = 10 cm

Hypotenuse = 10 cm

Example 2: In a right-angled triangle, the hypotenuse is 13 cm and one leg is 5 cm. Find the other leg.

Solution: Given: c = 13 cm, a = 5 cm, b = ?

Rearrange Pythagoras: b² = c² − a² = 13² − 5² = 169 − 25 = 144

b = √144 = 12 cm

Example 3: A ladder 17 m long is placed against a wall. The foot of the ladder is 8 m away from the base of the wall. At what height does the top of the ladder touch the wall?

Solution: Given, Hypotenuse = length of ladder = 17 m, Base = 8 m

Apply Pythagoras' theorem: height² = 17² − 8² = 289 − 64 = 225

height = √225 = 15 m

The ladder touches the wall at a height of 15 m

Example 4: A triangle has sides 7 cm, 24 cm, and 25 cm. Is it a right-angled triangle?

Solution: The largest side is 25 cm, this would be the hypotenuse if the given triangle is a right triangle.

Check the Pythagorean theorem:

7² + 24² = 25²

49 + 576 = 625

625 = 625

Since a² + b² = c², this is a right-angled triangle. (7-24-25 is a Pythagorean triple.)

Therefore, the given triangle is a right-angled triangle with the right angle opposite the 25 cm side.

Example 5: A ladder is placed against a wall to reach a window. The length of the ladder is 10 m and the distance between the foot of the ladder and the wall is 6 m. How high is the window from the ground?

Solution: Here, the length of the ladder is AC = 10 m, and the

The distance between the foot of the ladder and the wall is 6 m.

The ΔABC is right-angled at B.

AB² + BC² = AC² (by the Pythagoras theorem)

62+ BC² = 102

36 + BC2 = 100 ⇒ BC2 = 100 - 36

BC2 = 64 = 82

⸫ BC = 8 m (by taking square roots of both sides)

Therefore, the window is at a height of 8 m.

Practice Questions on Right Angled Triangle

1. ΔABC is right-angled at B. If AB = 5cm and BC =12cm, find the length of AC.

2. A triangle has sides of length 8 cm, 6 cm and 10 cm. Determine whether this is a right-angled triangle.

3. A right triangle has legs of 15 cm and 20 cm. Find (i) the hypotenuse and (ii) the area.

4. A pole of height 12 m stands vertically on the ground. A wire is tied from the top of the pole to a point 9 m from the base of the pole on the ground. Find the length of the wire.

5. Check whether a triangle with sides 11 cm, 60 cm, and 61 cm is a right-angled triangle.

6. The area of a right-angled triangle is 240 cm² and one leg is 30 cm. Find the other leg and the hypotenuse.

7. A tree is broken due to a storm. It broke at a height of 7 m and the top of the tree touches the ground at a distance of 24 m from the base of the tree. Find the actual height of the tree.

8. The hypotenuse of a right triangle is 26 cm. One leg is 10 cm. Find the length of the other leg.

9. Check whether a triangle with sides 7 cm, 24 cm, and 25 cm is a right-angled triangle.

10. Find the hypotenuse of a right triangle with base 9 cm and height 40 cm.