Long Question 3


Question 3:
In diagram below, PQRS is a quadrilateral. PTS and TUR are straight lines.
 
 
It is given that PQ =20 x ˜ ,   PT =8 y ˜ ,   SR =25 x ˜ 24 y ˜ ,   PT = 1 4 PS   and   TU = 3 5 TR
(a) Express in terms of x ˜ and/or   y ˜ :
 (i)   Q S
 (ii) T R
(b) Show that  the points  Q, U and S  are collinear.
(c) If   | x ˜ | = 2  and | y ˜ | = 3, find   | Q S |


Solution:

(a)(i)
QS = QP + PS QS =20 x ˜ +32 y ˜ Given  PT = 1 4 PS PS =4 PT =4( 8 y ˜ )=32 y ˜


(a)(ii)
T R = T S + S R T R = 3 4 P S + 25 x ˜ 24 y ˜ T R = 3 4 ( 32 y ˜ ) + 25 x ˜ 24 y ˜ T R = 24 y ˜ + 25 x ˜ 24 y ˜ T R = 25 x ˜


(b)
QU = QP + PT + TU QU =20 x ˜ +8 y ˜ + 3 5 ( 25 x ˜ ) Given TU = 3 5 TR QU =20 x ˜ +8 y ˜ +15 x ˜ QU =5 x ˜ +8 y ˜ From (a)(i)  QS =20 x ˜ +32 y ˜ QS QU = 20 x ˜ +32 y ˜ 5 x ˜ +8 y ˜ QS QU = 4( 5 x ˜ +8 y ˜ ) ( 5 x ˜ +8 y ˜ ) QS QU =4 QS =4 QU  Q, U and S are collinear.


(c)









P S = 32 y ˜ | P S | = 32 | y ˜ | | P S | = 32 × 3 = 96 P Q = 20 x ˜ | P Q | = 20 | x ˜ | | P Q | = 20 × 2 = 40 | Q S | = 96 2 + 40 2 | Q S | = 104