Question

167, If the perimeter of a right-angled triangle is 56 cm
and area of the triangle is 84 sq. cm, then the
length of the hypotenuse is (in cm)
(1) 25
(2) 50
(3) 7
(4) 24​

Answer 1

Answer:

option 1) 25 is the answer

Answer 2

The length of the hypotenuse is 25 cm

Solution:

Let "a", "b" be the legs of triangle

Let "c" be the length of hypotenuse

The perimeter of a right-angled triangle is 56 cm

Therefore,

a + b + c = 56

a + b = 56 - c

Area of the triangle is 84 sq. cm

$Area = \frac{1}{2} \times base \times height\\\\84 = \frac{1}{2} \times a \times b\\\\ab = 168$

By pythogoras theorem,

$c^2 = a^2 + b^2$

Rewriting we get,

$c^2 = (a+b)^2-2ab$

Substitute ab = 168

$c^2 = (a+b)^2-2(168)\\\\c^2 = (a+b)^2-336$

Substitute a + b = 56 - c

$c^2 = (56-c)^2 - 336\\\\c^2 = 3136 -112c + c^2 - 336\\\\-112c + 2800 = 0\\\\112c = 2800\\\\c = 25$

Thus the length of the hypotenuse is 25 cm

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