Question

Q52
Please help me with this question as soon as possible.

Answer 1

13.5 square units

Explanation:

The given circle with center Q has a circumference of 12π.

Using the formula C = 2πr for the circumference of a circle with radius r, we can find that the radius of circle Q:

`\begin{aligned}\sf Circumference&=2\pi r \\\\\implies 12 \pi&=2\pi r\\\\(12 \pi)/(2 \pi)&=(2\pi r)/(2\pi)\\\\6&=r\\\\r&=6\; \sf units\end{aligned}`

Since points R and S are on the circumference of circle Q, then QR and QS are radii of the circle. As r = 6, then:

`\overline{QR}=\overline{QS}=6`

To find the area of the shaded region, we can subtract the area of the smaller right triangle AQB from the area of the larger right triangle RQS.

The area of a triangle is half the product of its base and height.

Therefore, the area of the larger triangle RQS is:

`\begin{aligned}\textsf{Area\;of\;$\triangle RQS$}&=(1)/(2) \cdot \overline{QR} \cdot \overline{QS}\\\\&=(1)/(2) \cdot 6 \cdot 6\\\\&=3 \cdot 6\\\\& = 18\; \sf square\;units\end{aligned}`

As A is the midpoint of
`\overline{QR}` and B is the midpoint of
`\overline{QS}`, then:

`\overline{QA}=\overline{QB}=3`

Therefore, the area of the smaller triangle AQB is:

`\begin{aligned}\textsf{Area\;of\;$\triangle AQB$}&=(1)/(2) \cdot \overline{QA} \cdot \overline{QB}\\\\&=(1)/(2) \cdot 3 \cdot 3\\\\&=1.5 \cdot 3\\\\& = 4.5\; \sf square\;units\end{aligned}`

The area of the shaded region is the area of the smaller right triangle AQB subtracted from the area of the larger right triangle RQS:

`\begin{aligned}\textsf{Area\;of\;shaded\;region}&=\textsf{Area\;of}\;\triangle RQS - \textsf{Area\;of}\;\triangle AQB\\\\& = 18-4.5\\\\&=13.5\; \sf square\;units\end{aligned}`

Therefore, the area of the shaded region is 13.5 square units.