3.5 Coordinates and ratios - Cardiff Tutor Company

Question 1(a): [3 marks]

Point $D$ lies on a line $AB$ such that $AD$ is $\dfrac{2}{3}$ of the total length of $AB$ .

Given that $A(2, 4)$ and $B(17, 13)$ , find the $x$ and $y$ coordinates of the point $D$ .

Answer type: Multiple answers type 1

ANSWER: $x = 12$ , $y = 10$ .

WORKING:

$17 - 2 = 15$ ; $13 - 4 = 9$

$15 \times \dfrac{2}{3} = 10$ ; $9 \times \dfrac{2}{3} = 6$

$2 + 10 = 12$ ; $4 + 6 = 10$

$(12, 10)$

Question 1(b): [3 marks]

Point $E$ lies on a line $AB$ such that $AD$ is $\dfrac{5}{7}$ of the total length of $AB$ .

Given that $A(0, 10)$ and $B(-21, -25)$ , find the $x$ and $y$ coordinates of the point $D$ .

Answer type: Multiple answers type 1

ANSWER: $x = - 15$ , $y = - 15$ .

WORKING:

$-21 - 0 = - 21$ ; $-25 -10 = -35$

$-21 \times \dfrac{5}{7} = - 15$ ; $- 35 \times \dfrac{5}{7} = - 25$

$0 + (-15) = -15$ ; $10 + (-25) = - 15$

$(- 15, - 15)$

Question 2:

Point $C$ lies on the line segment $AB$ . $C$ is such that the ratio $AC:CB = 1:2$ .

Question 2(a): [2 marks]

What are the $x$ and $y$ coordinates of the point $C$ ?

Answer type: Multiple answers type 1

ANSWER: $x = - 2$ , $y = -2$ .

WORKING:

$A(-4,-5)$ , $B(2,4)$

$2 - (- 4) = 6$ ; $4 - (-5) = 9$

$6 \times \dfrac{1}{3} = 2$ ; $9 \times \dfrac{1}{3} = 3$

$- 4 + 2 = - 2$ ; $- 5 + 3 = - 2$

$( -2, -2)$

Question 2(b): [2 marks]

Another point $X$ lies on the line segment $AB$ .

This divides the line segment in the ratio $AC:CX:XB = 2:1:3$

What are the $x$ and $y$ coordinates of the point $X$ ?

Answer type: Multiple answers type 1

ANSWER: $x = - 1$ , $y = - 0.5$ .

WORKING:

$C(-2,-2)$ , $B(2,4)$

$CX:XB = 1:3$

$-2 - 2 = -4$ ; $-2 - 4 = -6$

$4 \times \dfrac{1}{4} = 1$ ; $6 \times \dfrac{1}{4} = 1.5$

$-2 + 1 = -1$ ; $-2 + 1.5 = -0.5$

$(-1 , -0.5)$

Question 3:

$F$ lies on a line $AB$ such that $AF$ is $\dfrac{3}{4}$ of the length of $AB$ .

Question 3(a): [2 marks]

Given that $A(0,0)$ and $B(20,-8)$ , find the $x$ and $y$ coordinates of the point $F$ .

Answer type: Multiple answers type 1

ANSWER: $x = 15$ , $y = - 6$

WORKING:

Change in $x = +20$ , Change in $y = -8$ .

$20 \times \dfrac{3}{4} = 15$ ; $- 9 \times \dfrac{3}{4} = - 6$

$(15, - 6)$

Question 3(b): [1 mark]

$G$ lies on $AF$ such that $AG:GF=1:2$ .

Write down the ratio $AG:GF:FB$ , in terms of $a:b:c$ .

Answer type: Multiple answers type 1

ANSWER: $a = 1, \, b = 2, \, c = 1$

Question 3(c): [2 marks]

Hence find the $x$ and $y$ coordinates of $G$ .

Answer type: Multiple answers type 1

ANSWER: $x = 5$ , $y = - 2$

WORKING:

$A(0,0)$ and $F(15,-6)$

Change in $x = + 15$ ; Change in $y = - 6$ .

$15 \times \dfrac{1}{3} = 5$ ; $- 6 \times \dfrac{1}{3} = - 2$

$(5, - 2)$

Question 4: [2 marks]

$X$ is a point on the line segment $AB$ .

$Y$ is a point on the line segment $CD$ .

$AX:XB = CY:YD = 1:2$ .

Which of the following graphs is the correct representation of the line $XY$ .

Answer type: Multiple choice type 2

ANSWER:

WORKING:

$X(5, -3)$ ; $Y(1, -7)$

$y \text{intercept} = - 8$

$\text{Gradient} = 1$

$y = x - 8$

Question 5:

The point $R$ lies on the line segment $PQ$ such that the ratio $PR:RQ = 1:4$

The point $T$ lies on the line segment $QS$ such that the ratio $QT:TS = 3:1$

The point $U$ lies on the line segment $TR$ such that the ratio $TU:UR = 1:2$

Given that $P = (2,1)$ , $Q = (12,6)$ and $S = (0,10)$ :

Question 5(a): [3 marks]

Find the coordinates of the point $R$

Answer type: Multiple choice type 1

A: $(4,2)$

B: $(2,4)$

C: $(10,5)$

D: $(5,10)$

ANSWER: A: $(4,2)$

WORKING:

In a ratio of $1:4$ there are $5$ parts in total, and the distance from $P$ to $R$ constitutes $1$ of those parts. Therefore, the distance from $P$ to $R$ counts for $\frac{1}{5}$ of the total distance between $P$ and $Q$ .

First, the $x$ coordinates: $12-2=10$ , then

$\frac{1}{5}\times 10 = 2$

Adding this to the $x$ coordinate of $P$ , gives the $x$ coordinate of $R$ :
$2+2=4$

Second, the $y$ coordinates: $6-1=5$ , then

$\frac{1}{5}\times 5=1$

Adding this to the $y$ coordinate of $P$ , gives the $y$ coordinate of $R$ :
$1+1=2$

Therefore, the coordinates of $R$ are $(4,2)$

Question 5(b): [3 marks]

Find the coordinates of the point $T$

Answer type: Multiple choice type 1

A: $(3,9)$

B: $(9,3)$

C: $(9,7)$

D: $(-2,-6)$

ANSWER: A: $(3,9)$

WORKING:

In a ratio of $3:1$ there are $4$ parts in total, and the distance from $Q$ to $T$ constitutes $3$ of those parts. Therefore, the distance from $Q$ to $T$ counts for $\frac{3}{4}$ of the total distance between $Q$ and $S$ .

First, the $x$ coordinates: $0-12=-12$ , then

$\frac{3}{4}\times (-12) = -9$

Adding this to the $x$ coordinate of $Q$ , gives the $x$ coordinate of $T$ :
$12-9=3$

Second, the $y$ coordinates: $10-6=4$ , then

$\frac{3}{4}\times 4=3$

Adding this to the $y$ coordinate of $Q$ , gives the $y$ coordinate of $T$ :
$6+3=9$

Therefore, the coordinates of $T$ are $(3,9)$

Question 5(c): [3 marks]

Find the coordinates of the point $U$

Answer type: Multiple choice type 1

A: $\left(\dfrac{10}{3}, \dfrac{20}{3} \right)$

B: $\left(\dfrac{23}{4}, \dfrac{13}{4} \right)$

C: $(3,9)$

D: $(4,2)$

ANSWER: A: $\left(\dfrac{10}{3}, \dfrac{20}{3} \right)$

WORKING:

In a ratio of $1:2$ there are $3$ parts in total, and the distance from $T$ to $U$ constitutes $1$ of those parts. Therefore, the distance from $T$ to $U$ counts for $\frac{1}{3}$ of the total distance between $T$ and $R$ .

First, the $x$ coordinates: $4-3=1$ , then

$\frac{1}{3}\times (1) = \frac{1}{3}$

Adding this to the $x$ coordinate of $T$ , gives the $x$ coordinate of $U$ :
$3 + \frac{1}{3} = \frac{10}{3}$

Second, the $y$ coordinates: $2-9=-7$ , then

$\frac{1}{3}\times (-7)= - \frac{7}{3}$

Adding this to the $y$ coordinate of $T$ , gives the $y$ coordinate of $U$ :
$9 - \frac{7}{3} = \frac{20}{3}$

Therefore, the coordinates of $T$ are $\left(\dfrac{10}{3}, \dfrac{20}{3} \right)$

Question 6: [4 marks]

Point $B$ lies on a line $AC$ such that $AB:BC = 3:2$

Given that $A = (3,q)$ , $B = (9,4q)$ and $C = (p,6)$ , find the values of $p$ and $q$

Answer type: Multiple answers type 1

ANSWER:

p = 13

q = 1

WORKING:

In a ratio of $3:2$ there are $5$ parts in total, and the distance from $A$ to $B$ constitutes $3$ of those parts. Therefore, the distance from $A$ to $B$ counts for $\frac{3}{5}$ of the total distance between $A$ and $C$ .

First, the $x$ coordinates of $B$ :

$p-3$ , then

\dfrac{3}{5}\times (p-3) = \dfrac{3p-9}{5}

Adding this to the $x$ coordinate of $A$ , gives the $x$ coordinate of $B$ :
$3 + \dfrac{3p-9}{5} = \dfrac{3p+6}{5}$

We know that the $x$ coordinate of $B$ is $9$ , therefore we can form an equation and solve for $p$ :

$\dfrac{3p+6}{5} = 9$

$3p+6=45$

$3p=39$

$p=13$

Second, the $y$ coordinates of $B$ :

$6-q$ , then

\dfrac{3}{5}\times (6-q)= \dfrac{18-3q}{5}

Adding this to the $y$ coordinate of $A$ , gives the $y$ coordinate of $B$ :
$q + \dfrac{18-3q}{5} = \dfrac{18+2q}{5}$

We know that the $y$ coordinate of $B$ is $4q$ , therefore we can form an equation and solve for $q$ :

$\dfrac{18+2q}{5}=4q$

$18+2q=20q$

$18q=18$

$q=1$