Question

the length of the sides of the triangle are in the ratio 4 is to 5 is 26 and its perimeter is 150 cm find the area of the triangle and the height corresponding to the longest side
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Answer 1

Question-

The length of the sides of the triangle are in the ratio 4 : 5 : 6 and its perimeter is 150 cm. Find the area of the triangle and the height corresponding to the longest side.

Answer-

Let the unknown part of the side be x. Then,

4x + 5x + 6x = 150cm

⇒ 15x= 150cm

⇒ x = 150/15 cm

⇒ x = 10 cm

• 4x = 4 × 10 = 40 cm
• 5x = 5 × 10 = 50 cm
• 6x = 6 × 10 = 60 cm

s = a + b + c ÷ 2 = 40 + 50 + 60 ÷ 2 = 150 ÷ 2 = 75 cm

Area using Heron's formulae-

√s(s - a)(s - b)(s - c) cm²

= √75(75 - 40)(75 - 50)(75 - 60) cm²

= √75 × 35 × 25 × 15 cm²

= √(3 × 5 × 5) × (5 × 7) × (5 × 5) × (3 × 5) cm²

= 3 × 5 × 5 × 5√7 cm²

= 375√7 cm²

The longest side = 60 cm

Unknown height corresponding to the longest side = h

Then,

Area = 1/2 × base × height

⇒ 375√7 cm²= 1/2 × 60 cm × h

⇒ 375√7 cm²= 30h

⇒ h = 375√7 cm²/30 cm

⇒ h = 25√7/2 cm

Area = 375√7 cm²

Height corresponding to the longest side = 25√7/2