Question

The base of an isosceles triangle 24cm and its area is 192squar cm . Find its perimeter

Answer 1

$\large{\underline{\bf{\purple{Given:-}}}}$

• ✦ Base of isosceles triangle ⠀⠀⠀⠀⠀= 24cm

• ✦ Are of triangle = 192 cm²

$\large{\underline{\bf{\purple{To\:Find:-}}}}$

• ✦ we need to find the perimeter of triangle.

$\huge{\underline{\bf{\red{Solution:-}}}}$

Let the side of triangle be a cm.

Then,

$\leadsto \rm{\pink{\:area = \frac{1}{4}b \sqrt{4 {a}^{2} - {b}^{2}}} } \: \: \: sq \: units \\ \\ \leadsto \rm\: \frac{1}{4} \times 24 \times \sqrt{4 {a}^{2} - 576 } \: \: {cm}^{2} \\ \\ \leadsto \rm\:12 \times \sqrt{ {a}^{2} - 144 } \: \: {cm}^{2}$

• ✦ But it is given that Area = 192cm²

⠀⠀

$\leadsto \rm\:\: \therefore \: \: \: \: 12 \times \sqrt{ {a}^{2} - 144} = 192 \\ \\\leadsto \rm\: \sqrt{ {a}^{2} - 144 } = { \cancel{\frac{192}{16}}} \\ \\\leadsto \rm\:\: \sqrt{ {a}^{2} - 144 } = 16 \\ \\\leadsto \rm\:\: {a}^{2} - 144 = {16}^{2} \\ \\ \leadsto \rm\:\: {a}^{2} - 144 = 256 \\ \\ \leadsto \rm\:\: {a}^{2} = 256 + 144 \\ \\ \leadsto \rm\:\: {a}^{2} =400 \\ \\ \leadsto \rm\:\:a = \sqrt{400} \\ \\\leadsto \rm\:\:a = 20$

• ✦ So perimeter of triangle

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀=(2a+b)cm

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀=(2 × 20 + 24)cm

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀= 64cm.

✦ perimeter of triangle = 64cm.

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Answer 2

Step-by-step explanation:

Given -

• Base of an isosceles triangle is 24 cm
• Area = 192 cm²

To Find -

• Perimeter of an isosceles triangle

As we know that :-

• Area of isocelses triangle = 1/4×b×√4a²-b²

→ 192 = 1/4 × 24 × √4a² - (24)²

→ 192 = 6 × √4a² - 576

→ 32 = √4a² - 576

→ (32)² = 4a² - 576

→ 1024 + 576 = 4a²

→ 1600 = 4a²

→ a² = 400

→ a = √400

→ a = 20 cm

As we know that :-

Perimeter of isocelses triangle = (2a + b)

→ (2×20 + 24)

→ 40 + 24

→ 64 cm

Hence,

The perimeter of an isosceles triangle is 64 cm.