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If Two Sides of a Triangle Are Unequal, Then the Angle Opposite to the Greater Side Is Greater Than Angle Opposite to the Smaller Side

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Question

In triangle, $∠ B = 90^{\circ}$ , $A B = (2 x + 1)$ cm and $B C = (x + 1)$ cm . If the area of the triangle ABC is $60 c m^{2}$ ,find its perimeter .

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Solution

Area of triangle $= \frac{1}{2} \times$ base $\times$ height
$60 = (\frac{1}{2}) \times (2 x + 1) \times (x + 1) 120 = (2 x + 1) (x + 1) 120 = 2 x ² + 2 x + x + 1 120 = 2 x ² + 3 x + 1 2 x ² + 3 x - 119 = 0 2 x ² + 17 x - 14 x - 119 = 0 x (2 x + 17) - 7 (2 x + 17) = 0 (2 x + 17) (x - 7) = 0$
Therefore, $x = 7$ as $x$ cant be negative.
$A B = 7 \times 2 + 1 = 15 c m$
$B C = 7 + 1 = 8 c m$
As given triangle is a right angled triangle,
so hypotenuse $= \sqrt{(15)^{2} + (8)^{2}}$
$= \sqrt{289}$
$= 17$ cm
Therefore, perimeter $= 15 + 8 + 17$
$= 40$ cm

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