Find length of line and the area of triangle

Question

Can anyone help me on this problem (attach)??

I learn this topic since I study at home. I don’t know how to figure out this problem. This is not homework, I just try to find the solution.

Here i find properties for similar triangle :

1. AAA (angle angle angle) All three pairs of corresponding angles are the same.
2. SSS in same proportion (side side side) All three pairs of corresponding sides are in the same proportion.
3. SAS (side angle side) Two pairs of sides in the same proportion and the included angle equal

Solution for (b) i. I got RX = 5.8.

The equation for RX SX/QX = XR/XP

Please advice

Answer 1

Do you know that angles on same arc are same?? Use this to prove (a) and (b)(i) follows directly from the previous result.For (b) (ii) Use the fact that ratio of areas of triangles sharing same altitude is equal to ratio of bases

Answer 2

(a) $\angle PXQ = \angle SXR$, $\angle XPQ = \angle XRS$, $\angle PQX = \angle RSX$, which leads to the similarity of the triangles $PXQ$ and $SXR$.

(b.i) Use the chord theorem to get $PX\cdot RX= SX\cdot QX$, which leads to $RX= \frac{4.5 \cdot 6.3}{8.1}=3.5$.

(b.ii) $\frac{Area_{PXQ}}{Area_{PSX}}= \frac{XQ}{XS}=\frac{6.3}{4.5}=\frac75$.