Question

Find the area of the right triangle whose one side passes through the center of the circle having a radius of $5\text{units}$ and base of the triangle lies on a chord of length $8\text{cm}$.

• A. $8{\text{unit}}^2$
• B. $16{\text{unit}}^2$

Answer 1

The correct option is B $16{\text{unit}}^2$

The given figure is as follows:

P is the center of the circle.
$PZ=5\text{units}$ (radius)

The given triangle is right triangle.
$\therefore \angle XQY={90}^o$

Hence, PQ bisects the chord ZY at the midpoint Q.
$\Rightarrow ZQ=QY=\frac{8}{2}=4\text{units}$

For $△PZQ,$
$P{Z}^2=Z{Q}^2+P{Q}^2$
$\Rightarrow P{Q}^2=P{Z}^2-Z{Q}^2=5^2-4^2=3^2$
$\Rightarrow PQ=3\text{units}$

Now, XP is also the radius of the circle.
$\therefore XQ=XP+PQ=5+3=8\text{units}$

Area of $△XYQ$
$=\frac{1}{2}\times XQ\times QY$
$=\frac{1}{2}\times 8\times 4{\text{unit}}^2$
$=16{\text{unit}}^2$