**Question 11:**

The Town Council plans to build an equilateral triangle platform in the middle of a roundabout. The diameter of circle

*RST*is 24 m and the perpendicular distance from

*R*to the line

*ST*is 18 m. as shown in Diagram below.

Find the perimeter of the platform.

*Solution*:Given diameter = 24 m

hence radius = 12 m

*O*is the centre of the circle.

Using Pythagoras’ theorem:

$\begin{array}{l}{x}^{2}={12}^{2}-{6}^{2}\\ x=\sqrt{144-36}\\ \text{}=10.39\text{m}\\ TS=RS=RT\\ \text{}=10.39\text{m}\times 2\\ \text{}=20.78\text{m}\\ \text{Perimeteroftheplatform}\\ TS+RS+RT\\ =20.78\times 3\\ =63.34\text{m}\end{array}$

**Question 12:**

Amy will place a ball on top of a pillar in Diagram below. Table below shows the diameters of three balls

*X*,

*Y*and

*Z*.

Which ball

*X*,

*Y*or

*Z*, can fit perfectly on the top of the pillar? Show the calculation to support Amy’s choice.

*Solution*:$\begin{array}{l}\text{Lettheradiusofthetopofthepillar}=r\text{cm}\text{.}\\ O\text{isthecentreofthecircle}\text{.}\\ \text{In}\Delta \text{}OQR,\\ {r}^{2}={\left(r-4\right)}^{2}+{8}^{2}\text{}\left(\text{usingPythagoras'theorem}\right)\\ {r}^{2}={r}^{2}-8r+16+64\\ {r}^{2}={r}^{2}-8r+80\\ {r}^{2}-{r}^{2}+8r=80\\ 8r=80\\ r=\frac{80}{8}\\ r=10\text{cm}\\ \\ \text{Therefore,diameter}\\ =2\times 10\\ =20\text{cm}\\ \\ \text{Ball}Y\text{withdiameter20cmcanfitperfectly}\\ \text{ontopofthepillar}\text{.}\end{array}$

**Question 13:**

Diagram below shows a rim of a bicycle wheel with a diameter of 26 cm. Kenny intends to build a holder for the rim.

Which of the rim holder,

*X*,

*Y*or

*Z*, can fit the bicycle rim perfectly? Show the calculation to support your answer.

*Solution*:$\begin{array}{l}\text{Lettheradiusoftherimholder}=r\text{cm}\text{.}\\ O\text{isthecentreofthecircle}\text{.}\\ \text{In}\Delta \text{}OQR,\\ {r}^{2}={\left(r-8\right)}^{2}+{12}^{2}\text{}\left(\text{usingPythagoras'theorem}\right)\\ {r}^{2}={r}^{2}-16r+64+144\\ {r}^{2}={r}^{2}-16r+208\\ {r}^{2}-{r}^{2}+16r=208\\ 16r=208\\ r=\frac{208}{16}\\ r=13\text{cm}\\ \\ \text{Therefore,diameter}\\ =2\times 13\\ =26\text{cm}\\ \\ \text{Rimholder}Z\text{withdiameter26cmcanfitthebicycleperfectly}\text{.}\end{array}$