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Area of a Right Triangle:

A right triangle is a triangle that has one of its angles a right angle. A triangle is a regular polygon with three sides, the sum of two sides being greater than the other. The sum of all the angles is 1800.  The name Right angled triangle is generally used in British English, while Americans call it a right triangle. Let us look at the formulas for obtaining the area of a right triangle.

Image Credits: Wikipedia

The side of the triangle containing the right angle are called legs of the triangle, or catheti, from the singular cathetus. The side opposite to the right angle is called Hypotenuse. The right triangle and the relationship with its sides form the basis of trigonometry. A Pythagorean triangle is one in which the lengths of all the three sides are integers.

There may be different types of right triangles like general triangles, isosceles triangle, 30-60-90 triangle etc. In an isosceles triangle, the two angles are 45 degrees, whereas in the other case the angles are 30 degrees and 60 degrees.

Length of the hypotenuse:

If a and b are the base lengths of the triangle, hypotenuse c is calculated by

C2 = a2 + b

This is according to Pythagoras’s theorem , which states “In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).”

Area of a right triangle

The area of a right triangle is found by the formula = ½ * b *h

The area of a right triangle with sides a,b and hypotenuse c is :

A = ½ * a * b

Alternatively, another formula is Heron’s formula, which states

A= √(s(s – a) (s – b) (s – c))

Where S = (a + b + c)/2

Examples

Question 1: Find the length of the hypotenuse of the right triangle if the length of the other two sides is 5 cm and 6 cm. Also, calculate the area and perimeter of the triangle.
Solution:

Given,
a = 5 cm
b = 6 cm

using the Pythagoras formula,
c2 = a2 + b2
c2 = (5 cm)2 + (6 cm)2
c2 = 25 cm2 + 36 cm2
c2 = 61 cm2
c = 7.81 cm

Area of the right triangle
= ½ *a * b
= 1/2 * 5 cm * 6 cm
= 15 cm2

Perimeter of the right triangle
= a + b + c
= 5 cm + 6 cm + 7.81 cm
= 18.8 cm

Question 2: Find the area of a right angled triangle whose hypotenuse is 15 cm and one of the sides is 12 cm.
Solution: 

AB² = AC² – BC²

= 15² – 12²

= 225 – 144

= 81

Therefore, AB = 9

Therefore, the area of the triangle = ¹/₂ × base × height

= ¹/₂ × 12 × 9

= 54 cm²

Question 3: The base and height of the triangle are in the ratio 3 : 2. If the area of the triangle is 243 cm² find the base and height of the triangle.
Solution: 

Let the common ratio be x

Then height of triangle = 2x

And the base of triangle = 3x

Area of triangle = 243 cm²

Area of triangle = 1/2 × b × h 243 = 1/2 × 3x × 2x

⇒ 3x² = 243

⇒ x² = 243/3

⇒ x = √81

⇒ x = √(9 × 9)

⇒ x = √9

Therefore, height of triangle = 2 × 9

= 18 cm

Base of triangle = 3x

= 3 × 9

= 27 cm
Question 4: Find the area of a triangle whose sides are 41 cm, 28 cm, 15 cm. Also, find the length of the altitude corresponding to the largest side of the triangle.
Solution: 

Semi-perimeter of the triangle = (a + b + c)/2

= (41 + 28 + 15)/2

= 84/2

= 42 cm

Therefore, area of the triangle = √(s(s – a) (s – b) (s – c))

= √(42 (42 – 41) (42 – 28) (42 – 15)) cm²

= √(42 × 1 × 27 × 14) cm²

= √(3 × 3 × 3 × 3 × 2 × 2 × 7 × 7) cm²

= 3 × 3 × 2 × 7 cm²

= 126 cm²

Now, area of triangle = 1/2 × b × h

Therefore, h = 2A/b

= (2 × 126)/41

= 252/41

= 6.1 cm

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