Question

In Fig. ABC is a right-angled traingle, $\angle B={90}^{\circ }$ AB = 28 cm and BC = 21 cm. With AC as diameter a semicircle is drawn and with BC as radius a quarter circle is drawn. Find the area of the shaded region correct to two decimal places.

Answer 1

votes in Math, Class XI-Science

Area of the shaded region

= Area of the semi-circle with AC as diameter + Area of ΔABC – Area of quadrant of circle with BC as radius

In right ΔABC,

$A{C}^2=A{B}^2+B{C}^2$

$A{C}^2={28}^2+{21}^2$

$A{C}^2=1225c{m}^2$

$AC=35cm$

∴ Area of the shaded region,

$=\frac{1}{2}\times \pi \times (\frac{AC}{2}{)}^2+\frac{1}{2}\times BC\times AB-\frac{1}{2}\times \pi \times (AC{)}^2$

$=\frac{1}{2}\times \frac{22}{7}\times (\frac{35}{2}{)}^2+\frac{1}{2}\times 21\times 28-\frac{1}{2}\times \frac{22}{7}\times (21{)}^2$

$=481.25+294-346.5=428.75c{m}^2$

Thus, the area of the shaded region is 428.75 cm2