Question

Angle ABC of triangle ABC is a Right angle. The sides of ABC are the diameters of semicircles as shown. The area of the semicircle on AB equals 8pi and the arc of the semicircle on AC has length 8.5pi. What is the radius of the semicircle of BC

Answer 1

7.5 Units

Explanation:

Angle ABC of triangle ABC is a Right angle. The sides of ABC are the diameters of semicircles

The area of the semicircle on AB equals 8pi

Area of a semicircle
`=(\pi r^2)/(2)`

Therefore:

`(\pi r^2)/(2)=8\pi\\r^2=16\\r=4`

Next, the arc of the semicircle on AC has length 8.5pi.

Length of arc of a semicircle =
`\pi r`

`\pi r=8.5\pi\\r=8.5`

Using Pythagoras theorem

`8.5^2=4^2+x^2\\x^2=8.5^2-4^2\\x^2=56.25\\x=√(56.25) \\x=7.5`

Radius of the semicircle of BC=7.5 Units