Question

Find the area of a traingle of sides 13cm,14cm,and 15cm.Also,Find the length of the perpendicular from the vertex opposite the side of length 14 cm.​

Answer 1

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Let, the sides of the triangle be a, b, c.

a = 13 cm

b = 14 cm

c = 15 cm

s = a+ b + c /2 = 13 + 14 + 15/2 = 42/2 = 21

we can find area of triangle by using 2 formulae

1. heron's formula
2. 1/2 × base × height

as we know sides of the triangle , we use heron's formula.

area of triangle =

$\sqrt{s(s - a)(s - b)(s - c)}$

$\sqrt{21(21 - 13)(21 - 14)(21 - 15)}$

$\sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056} = 84$

therefore, area of triangle = 84 cm sq.

( length of perpendicular = height of the triangle)

know we want the value of height of the triangle. so , use the 2nd formula to find height of the triangle.

we know ,

area of triangle = 84 cm sq.

base = 14 cm ( given in the question )

area of triangle = 1/2 × base × height

84 = 1/2 × 14 × height

84 = 7 × height

height = 84/7 = 12 cm.

therefore, length of the perpendicular = 12 cm.