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# The sides of a triangle are 35 cm, 54 cm and 61 cm, respectively. The length of its longest altitude

a. 16√5 cm

b. 10√5 cm

c. 24√5 cm

d. 28 cm

**Solution:**

Consider a triangle ABC

Given, the sides AB = 35 cm

BC = 54 cm

AC = 61 cm

We have to find the length of the longest altitude.

By __Heron’s formula__,

Area of triangle = √s(s - a)(s - b)(s - c)

Where s= semiperimeter

s = (a + b + c)/2

Now, s = (35 + 54 + 61)/2

= 150/2

s = 75 cm

Area of triangle = √75(75 - 35)(75 - 54)(75 - 61)

= √75(40)(21)(14)

= √15 × 5 × 8 × 5 × 7 × 3 × 7 × 2

= √5 × 3 × 5 × 4 × 2 × 5 × 7 × 3 × 7 × 2

= √5 × 5 × 3 × 3 × 4 × 7 × 7 × 2 × 2

= 5 × 3 × 2 × 7 × 2 × √5

= 5 × 4 × 21 × √5

= 20 × 21 × √5

= 420√5 cm²

Area of triangle = 1/2 × base × height

In triangle ABC,

Area of triangle = 1/2 × AB × CD

420√5 = 1/2 × 35 × CD

CD = (420√5 × 2)/35

CD = 12√5 × 2

CD = 24√5 cm

Therefore, the length of the longest altitude is 24√5 cm

**✦ Try This: **The sides of a triangle are 25 cm, 34 cm and 41 cm, respectively. The length of its longest altitude

**☛ Also Check: **NCERT Solutions for Class 9 Maths Chapter 12

**NCERT Exemplar Class 9 Maths Exercise 12.1 Problem 7**

## The sides of a triangle are 35 cm, 54 cm and 61 cm, respectively. The length of its longest altitude a. 16√5 cm, b. 10√5 cm, c. 24√5 cm, d. 28 cm

**Summary:**

The sides of a triangle are 35 cm, 54 cm and 61 cm, respectively. The length of its longest altitude is 24√5 cm

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