Question 6:
The point M (x, 4), is the midpoint of the line joining straight line Q (-2, -3) and R (14, y).
The value of x and y are

Solution:

$\begin{array}{l}x=\frac{-2+14}{2}\\ x=\frac{12}{2}\\ x=6\\ \\ 4=\frac{-3+y}{2}\\ 8=-3+y\\ y=11\end{array}$

Question 7:
In diagram below, PQR is a right-angled triangle. The sides QR and PQ are parallel to the y-axis and the x-axis respectively. The length of QR = 6 units.

Given that M is the midpoint of PR, then the coordinates of M are

Solution:
x-coordinate of R = 3
y-coordinate of R = 1 + 6 = 7
R = (3, 7)

$\begin{array}{l}P\left(1,1\right),R\left(3,7\right)\\ \text{Coordinates of }M\\ =\left(\frac{1+3}{2},\frac{1+7}{2}\right)\\ =\left(2,4\right)\end{array}$

Question 8:
Given points P (–2, 8) and Q (10, 8), find the length of PQ.

Solution:
$\begin{array}{l}\text{Length of }PQ\\ =\sqrt{{\left[10-\left(-2\right)\right]}^2+{\left(8-8\right)}^2}\\ =\sqrt{{\left(14\right)}^2+0}\\ =14\text{ units}\end{array}$

Question 9:
In diagram below, ABC is an isosceles triangle.

Find
(a) the value of k,
(b) the length of BC.

Solution:
$\begin{array}{l}\left(\text{a}\right)\\ \text{For an isosceles triangle, }\\ y-\text{coordinate of }C\text{ is the midpoint of straight line }AB.\\ \frac{2+k}{2}=-3\\ 2+k=-6\\ k=-8\\ \\ \left(\text{b}\right)\\ B=\left(-2,-8\right)\\ BC=\sqrt{{\left[10-\left(-2\right)\right]}^2+{\left[-3-\left(-8\right)\right]}^2}\\ =\sqrt{{12}^2+5^2}\\ =13\text{ units}\end{array}$

Question 10:
Diagram below shows a rhombus PQRS drawn on a Cartesian plane. PS is parallel to x-axis.

Given the perimeter of PQRS is 40 units, find the coordinates of point R.

Solution:
$\begin{array}{l}\text{All sides of rhombus have the same length,}\\ \text{therefore length of each side}=\frac{40}{4}=10\text{ units}\\ PQ=10\\ {\left(9-x_1\right)}^2+{\left(7-\left(-1\right)\right)}^2={10}^2\\ 81-18x_1+x_1{}^2+64=100\\ x_1{}^2-18x_1+45=0\\ \left(x_1-3\right)\left(x_1-15\right)=0\\ x_1=3,15\\ x_1=3\\ Q=\left(3,-1\right),R=\left(x_2,-1\right)\\ \\ QR=10\\ {\left(x_2-3\right)}^2+{\left[-1-\left(-1\right)\right]}^2={10}^2\\ x_2{}^2-6x_2+9+0=100\\ x_2{}^2-6x_2-91=0\\ \left(x_2+7\right)\left(x_2-13\right)=0\\ x_2=-7,13\\ x_2=13\\ \\ \therefore R=\left(13,-1\right)\end{array}$

Question 11:
Diagram below shows the route of a Negaraku Run.
The route of male participants is PQUT while the route for female participants is QRST. QR is parallel to UT whereas UQ is parallel to SR.
(a) Given the distance between point Q and point R is 9 km, state the coordinates of point R.
(b) It is given that the male participants start the run at point P and the female participants at point Q.
Find the difference of distance, in km, between the male and female participants.

Solution:
(a)
R = (9, 1)

(b)
Route of male participants
= PQ + QU + UT
= 10 + 7 + 7
= 24 km

Route of female participants
= QR + RS + ST
= 9 + 7 + 2
= 18 km

Difference in distance
= 24 – 18
= 6 km

Question 12:
Diagram below shows a parking lot in the shape of trapezium, RSTU. Two coordinates from the trapezium vertices are R(–30, –4) and S(20, –4).

(a) Given the distance between vertex S and vertex T is 22 units.
State the coordinates of vertex T.
(b) Given the area of the parking lot is 946 unit², find the coordinates of U.

Solution:
(a)
y-coordinates of T = –4 + 22 = 18  
T = (20, 18)

(b)
$\begin{array}{l}\text{Area of trapezium}=946{\text{ unit}}^2\\ \frac{1}{2}\times \left(UT+RS\right)\times ST=946\\ \frac{1}{2}\times \left(UT+50\right)\times 22=946\\ UT+50=\frac{946\times 2}{22}\\ UT+50=86\\ UT=36\\ \\ x-\text{coordinate of point }U=20-36=-16\\ y-\text{coordinate of point }U=18\\ U=\left(-16,18\right)\end{array}$

$\begin{array}{l}\frac{2+m}{2}=5\\ 2+m=10\\ m=8\end{array}$

$\begin{array}{l}P\left(-4,8\right),Q\left(6,-2\right)\\ \text{Coordinates of }M\\ =\left(\frac{-4+6}{2},\frac{8+\left(-2\right)}{2}\right)\\ =\left(1,3\right)\end{array}$

$\begin{array}{l}\text{Distance of }PQ\\ =\sqrt{{\left(-4-20\right)}^2+{\left[6-\left(-1\right)\right]}^2}\\ =\sqrt{{\left(-24\right)}^2+7^2}\\ =\sqrt{576+49}\\ =\sqrt{625}\\ =25\text{ units}\end{array}$

P (4, 3) and Q (10, 5).