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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