Question : The perimeter of an isosceles triangle is 544 cm and each of the equal sides is $\frac{5}{6}$ times the base. What is the area (in cm2) of the triangle?
Option 1: 38172
Option 2: 18372
Option 3: 31872
Option 4: 13872
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Correct Answer: 13872
Solution :
Let the base of the isosceles triangle as $b$ cm and each of the equal sides as $a$ cm.
Given that the perimeter of the triangle is $544$ cm.
$⇒2a + b = 544$ _____(i)
Given that each of the equal sides is $\frac{5}{6}$ times the base.
$⇒a = \frac{5}{6}b$ _____(ii)
Substituting $a$ in the equation (i),
$⇒2(\frac{5}{6}b) + b = 544$
$⇒\frac{5}{3}b + b = 544$
$⇒\frac{8}{3}b = 544$
$⇒b = 204$ cm
Substituting $b$ in the equation (ii),
$⇒a = \frac{5}{6}b$
$⇒a = 170$ cm
In an isosceles triangle, the height can be found using the Pythagorean theorem,
$⇒h = \sqrt{a^2 - (\frac{b}{2})^2}$
$⇒h = \sqrt{170^2 - (\frac{204}{2})^2}$
$⇒h = 136\;\operatorname{ cm }$
The area of the triangle $=\frac{1}{2}bh=\frac{1}{2} \times 204 \times 136 = 13872\operatorname{ cm^2 }$
Hence, the correct answer is 13872.
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