**Question 4**:

In the diagram above,

Calculate:

*AXB*is an arc of a circle centre*O*and radius 10 cm with ∠*AOB*= 0.82 radian.*AYB*is an arc of a circle centre*P*and radius 5 cm with ∠*APB*=*θ*.Calculate:

**(a)**the length of the chord

*AB*,

**(b)**the value of

*θ*in radians,

**(c)**the difference in length between the arcs

*AYB*and

*AXB*.

*Solution:***(a)**

$\begin{array}{l}\frac{1}{2}AB=\mathrm{sin}0.41\times 10\to \left(\text{ChangethecalculatortoRadmode}\right)\\ \frac{1}{2}AB=3.99\\ \therefore \text{Thelengthofchord}AB=3.99\times 2=7.98\text{}cm.\end{array}$

**(b)**

$\begin{array}{l}\text{Let}\frac{1}{2}\theta =\alpha ,\text{}\theta =2\alpha \\ \mathrm{sin}\alpha =\frac{3.99}{5}\\ \alpha =0.924\text{rad}\\ \therefore \theta =0.924\times 2=1.848\text{radian}\text{.}\end{array}$

**(c)**

Using

*s*=*rθ*Arcs

*AXB*= 10 × 0.41 = 4.1 cmArcs

*AYB*= 5 × 1.848 = 9.24 cmDifference in length between the arcs

*AYB*and*AXB*= 9.24 – 4.1

=

**5.14 cm**