Question

A wheel of radius $8$ units rolls along the diameter of a semicircle of radius $25$ units; it bumps into this semicircle. What is the length of the portion of the diameter that cannot be touched by the wheel?

• A. $12$
• B. $15$
• C. $17$
• D. $20$

Answer 1

The correct option is D $20$

Draw $\Delta OBC$, where $O$ is the centre of the large circle, $B$ is the centre of the wheel $C$ is the point of tangency of the wheel and the diameter of the semicircle and $BC$ is the radius of the wheel, so $\angle OCB={90}^o$.

We know that, if a line is tangent to the circle, it is perpendicular to the radius drawn to the point of tangency.

Thus $\Delta OBC$ is a right triangle.

We extend $OB$ to meet the semicircle at $D$.

Then $BD=BC=8$ (radius of the wheel)

Then, $OB=OD-BD=25-8=17$

Apply Pythagorean theorem to $\Delta OBC$

$\Rightarrow O{B}^2=O{C}^2+B{C}^2$

$\Rightarrow {17}^2=O{C}^2+8^2$

$\Rightarrow O{C}^2=225$

$\therefore OC=15$

Then $AC=x=OA-OC=25-15=10$

Thus length of the portion of the diameter that cannot be touched by the wheel is $2x=2\times 10=20units$