The hypotenuse of a right triangle is 6m more than twice the shortest side. If the third side is 2 m less than the hypotenuse, find the area (in m2) of the triangle.
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Correct answer: C
Explanation
The correct answer is option 3 i.e. 120.
Let the shortest side of the triangle be x meters.
Hypotenuse = 2x + 6 meters
Third side = (2x + 6) - 2 = 2x + 4 meters
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So, (2x + 6)2 = x2 + (2x + 4)2
⇒ 4x2 + 24x + 36 = x2 + 4x2 + 16x + 16
⇒ x2 - 8x - 20 = 0
⇒ (x - 10)(x + 2) = 0
⇒ x = 10 or x = -2
Since a side cannot be negative, x = 10
Shortest side = x = 10 m
Hypotenuse = (2x + 6) = 2(10) + 6 = 26 m
Third side = (2x + 4) = 2(10) + 4 = 24 m
In this case, we can consider the shortest side (10 m) as the base and the third side (24 m) as the height.
Area of a right triangle = (1/2) × Base × Height = (1/2) × 10 × 24 = 120 m2
Therefore, the area of the triangle is 120 m2.