Question 19

In the adjoining figure, ABC is a right-angled triangle. $$\angle ABC = 90^\circ$$, $$AB = 18$$ cm, $$BC = 24$$ cm. There are three squares drawn externally on the hypotenuse AC as shown, such that AC would be the sum of the side lengths of these squares. Area of two of these squares is $$225$$ $$\text{cm}^2$$ and $$64$$ $$\text{cm}^2$$ respectively, as indicated in the figure. Then the area of the shaded region = _______ $$cm^{2}$$

Correct Answer: 52 $$cm^2$$

Solution

By Pythagoras, $$AC = \sqrt{18^2 + 24^2} = \sqrt{324 + 576} = 30$$ cm. The two known squares have sides $$\sqrt{225} = 15$$ cm and $$\sqrt{64} = 8$$ cm, and the three side lengths add up to AC, so the third square has side $$30 - 15 - 8 = 7$$ cm. Its area, the shaded region, is $$7^2 = 49$$ $$\text{cm}^2$$.