IMAT 2016 Q53 [Hypotenuse]

A right­-angled triangle has an area of 18 cm². One of the two shorter sides is twice the length of the other one.

What is the length of the hypotenuse of the triangle?

A. 9\sqrt2 cm
B. 6\sqrt5 cm
C. 3\sqrt10 cm
D. 3\sqrt5 cm
E. 3\sqrt6 cm

In order to solve this question, we must first know what a right-angled triangle is. It’s a triangle in which one angle is a right angle (that is, a 90^{\circ}) or two sides are perpendicular.

Having that in mind, let’s draw a triangle with the information given.

From the text we know that a=2b.

The area of the triangle is calculated by: P = \frac{a \cdot b}{2} = 18 cm^2

Using the a=2b in the formula for area, we can get the value of the sides: P = \frac{2b \cdot b}{2} = \frac{2b^2 }{2} =b^2 = 18 cm^2

Knowing the Pytagoras Theorem c^2 = a^2 +b^2

in our case: c^2 = a^2 +b^2 = (2b)^2 + b^2 = 4b^2+b^2=5 b^2.

From the previously calculated, b^2 = 18 cm^2 \to c^2 = 5 \cdot 18 cm^2 = 5\cdot 2\cdot 9cm^2 \to c = \sqrt{5\cdot 2\cdot 9} cm
\to c = \sqrt{5\cdot 2}\cdot 3 cm \to c = 3\cdot\sqrt{5\cdot 2}cm = 3\cdot\sqrt{10} cm

So the correct answer is: (C)c=3\sqrt{10} cm