Question

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

A

165cm

B

105cm

C

245cm

D

28 cm

Solution

## The correct option is C 24√5cm Thinking Process (i) First, determine the semi-perimeter, s and then determine the area of the triangle by using Heron’s formula. (ii) For the longest altitude, take base as the smallest side. Apply the formula, Area=12×Base×Altitude (iii) Equate the area obtained using the two formulae and obtain the required height. Let ABC be a triangle in which sides AB = 35 cm, BC = 54 cm, CA = 61 cm Now, semi-perimeter of the triangle, s=a+b+c2=35+54+612=1502=75 cm [∵ semi−perimeter, s=a+b+c2] ∵    Area of ΔABC=√s(s−a)(s−b)(s−c) =√75(75−35)(75−54)(75−61) =√75×40×21×14 =√25×3×4×2×5×7×3×7×2 =5×2×2×3×7√5=420√5 cm2 Also  Area of ΔABC=12×AB×Altitude ⇒      12×35×CD=420√5 ⇒      CD=420×2√535 ∴      CD=24√5 Hence, the length of altitude is 24√5 cm

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