Question

the length of hypotenuse of a right angled triangle is 13 and the length of another side is 5 cm. Find its area.
using herons formula

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Answer 1

Step-by-step explanation:

H²=P²+B²

13²=5²+B²

169-25=B²

B²=144

B=√144

B=12cm

Let a=13, b=12, c=5

S = a+b+c/2

13+12+5/2

=15

Using herons formula = √s(s-a)(s-b)(s-c)

= √15(15-13)(15-12)(15-5)

= √15×2×3×10

= √3×5×2×3×2×5

=2×3×5

=30cm²

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Answer 2

Given -

• Length of hypotenuse = 13

• Length of another side = 5

To find -

• Area of the triangle.

Solution -

Here, we are given with the 2 sides of a triangle, but, for using heron's formula, we need to find the 3rd side too, for that first we will use Pythagoras theorem.

Let the 3rd side be termed as B

So,

Pythagoras theorem = H² = A² + B²

where,

H = Hypotenuse

A = 1st side

B = 2nd side

On substituting the values -

$\longmapsto$ (13)² = (5)² + (B)²

$\longmapsto$ 169 = 25 + B²

$\longmapsto$ B² = $\sf\sqrt{144}$

$\longmapsto$ B = 12

$\therefore$ The 3rd side is 12

Now,

We will find the area of the triangle, by using heron's formula. First we will find the semi-perimeter of the triangle then we will find the area.

So,

Semi-perimeter = $\sf\dfrac{a + b + c}{2}$

Semi-perimeter = $\sf\dfrac{13 + 5 + 12}{2}$

Semi-perimeter = $\sf\cancel\dfrac{30}{2}$

Semi-perimeter = 15

At the end -

Area of triangle = $\sf\sqrt{s(s - a)(s - b)(s - c)}$

On substituting the values -

$\sf\sqrt{15(15 - 13)(15 - 5)(15 - 12)}$

$\sf\sqrt{15 \times 2 \times 10 \times 3}$

$\sf\sqrt{900}$

$30 \: Ans$

$\therefore$ The area of triangle is 30

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