Question

The perimeter of a right angled triagle is 24 cm. If its hypotenuse is 10 cm then area of triangle

Answer 1

$<b>Heya mate. (^_-). Solution below.$
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• Given Perimeter of a triangle = 24 cm


• We know that perimeter of a ∆ = sum of all sides

=> 24 = first side + second side + third side

=> 24 = first side + second side + 10

=> 24 - 10 = first side + second side.

=> 14 = first side + second side.



• Now, Let the first side be 'x'.

• Then, second side = 14 - x



• Now, since it is a right angled triangle, we will use the Pythagoras' Theorem, according to which, square of hypotenuse = sum of square of other two sides.


=> [ (x)² + (14 - x)²] = (10)²


• Using identity : (a - b)² = a² + b² - 2ab -


=> [ (x²) + (14² + x² (- 2 × 14 × x))] = 100

=> x² + 196 + x² - 28x = 100


• Shifting 100 to LHS -


=> x² + 196 + x² - 28x - 100 = 0


• Rearranging the terms -

=> x² + x² - 28x + 196 - 100 = 0

=> 2x² - 28x + 96 = 0


• Taking 2 in common -

=> 2 [x² - 14x + 48] = 0


• Shifting 2 to RHS -

=> x² - 14 + 48 = 0 ÷ 2

=> x² - 14x + 48 = 0


• Using the method of midterm split -

=> x² - 8x - 6x + 48 = 0

=> x(x - 8) - 6(x - 8) = 0

=> (x - 6)(x - 8) = 0


• Using the zero product rule -

x = 6

or, x = 8




• Now, since 6 + 8 = 14, 6 cm and 8 cm is the measurement of first and second side respectively.

• So, first side = 6 cm

• Second side = 14 - 6 = 8 cm




• Now, area of a right angled triangle = half the product of the two sides of the triangle (leaving the hypotenuse)


• Then, area of the ∆

= ½ × 6 × 8

= 3 × 4

= 12 cm²


• Therefore, the area of the triangle = 12 cm².

$<marquee>Hence, your answer is 12 cm².</marquee>$
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Thank you... ;-)