Question

A semicircle is drawn with AB as its diameter. From C, a point on AB, a line perpendicular to AB is drawn, meeting the circumference of the semicircle at D. Given that AC=2cm and CD=6cm, the area of the semicircle in square cm will be ?

Answer 1

Given that

AC = 2 cm

CD = 6 cm

As angle in a semi circle is a right angle  

∠ADC=90°  

$CD^2$=AC×CB -------  (1)

Then we know the value of AC and CD to find the value of CB  

(1) becomes CB=$\frac{CD^2}{AC}$

  Substituting the value of AC and CD  

CB=$\frac{6^2}{2}$  

CB=$\frac{36}{2}$  

CB=18 cm  

AB=AC+CB  

AB=18+2=20 cm  

If the diameter of the semicircle is 20 cm then the radius of the semicircle be $r=\frac d2=\frac{20}2=10\ cm$  

Area of Semicircle = $\frac12\pi r^2$

                             = 12×$\pi$×10×10  

                            = 1×$\pi$×5×10

                           =50$\pi\ cm^2$

Answer 2

Answer:

Let O be the centre of the circle. Then, OA=OB=OD=r

Now, OC=r−2

and CD=6

Thus, in △ODC

OC

2

+CD

2

=OD

2

(r−2)

2

+6

2

=r

2

r

2

+4−4r+36=r

2

4r=40

r=10 cm

Area of semicircle =

2

πr

2

= 2

π(10)

2

=50π