A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 5: Straight line graphs · Lesson 2

Parallel and perpendicular lines, length and area

73 marks

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Date

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Quick questions to start.

A1
Needed today
Find the gradient of a line perpendicular to $2 x + 5 y - 3 = 0$ .
[2 marks]
Answer
A2
Last lesson
Find an equation of the line through $(- 2, 7)$ and $(4, - 5)$ in the form $y = m x + c$ .
[2 marks]
Answer $y = \dots\dots$
A3
From chapter 1
Write $6^{2} + 3^{2}$ in the form $a 5$ .
[2 marks]
Answer
A4
From chapter 2
Solve $(x - 3)^{2} + 16 = 41$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

For each pair, decide whether the lines are parallel, perpendicular or neither. (a) $4 x - 6 y + 5 = 0$ and $y = \frac{2}{3} x - 1$ . (b) $5 x + 2 y = 7$ and $2 x - 5 y + 1 = 0$ .

Hint Write each line as

y = m x + c

. Parallel: equal gradients. Perpendicular: the gradients multiply to

- 1

B1
The lines $l_{1}$ , $l_{2}$ and $l_{3}$ have equations $3 x + 7 y - 2 = 0$ , $14 x - 6 y + 9 = 0$ and $6 x + 14 y + 11 = 0$ . Which two lines are parallel? Which line is perpendicular to the other two?
[3 marks]
Answer

Example 2

Find an equation of the line through $(2, - 3)$ that is perpendicular to $3 x - 5 y + 4 = 0$ . Give it in the form $a x + b y + c = 0$ , where $a$ , $b$ and $c$ are integers.

Hint Find the gradient of the given line, then use

- \frac{1}{m}

B2
Find an equation of the line through $(- 1, 5)$ that is perpendicular to $4 x + 7 y - 2 = 0$ . Give it in the form $a x + b y + c = 0$ , where $a$ , $b$ and $c$ are integers.
[4 marks]
Answer

Example 3

The distance between $(2, p)$ and $(p, 8)$ is $25$ . Find the two possible values of $p$ .

Hint Square the distance:

(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} = 20

. Then solve the quadratic.

B3
The distance between $(q, 2)$ and $(4, q)$ is $213$ . Find the two possible values of $q$ .
[4 marks]
Answer $q = \dots\dots$ or $q = \dots\dots$

Example 4

The points $P (- 3, 0)$ , $Q (3, 3)$ and $R (1, 7)$ are the vertices of a triangle. Show that angle $P QR$ is a right angle, and find the area of the triangle.

Hint Two sides at

Q

are perpendicular if their gradients multiply to

- 1

. Then those two sides are the base and the height.

B4
The points $A (- 2, - 1)$ , $B (2, 1)$ and $C (1, 3)$ are the vertices of a triangle. Show that angle $A B C$ is a right angle, and find the area of the triangle.
[4 marks]
Answer

Practice

A-Level

C1
The points $A (- 4, 2)$ and $B (2, 5)$ lie on the line $l$ . The line $m$ passes through $B$ and is perpendicular to $l$ . Find the coordinates of the point where $m$ crosses the $x$ -axis.
[4 marks]
Answer $(\dots\dots, 0)$

C2
The lines $(k + 1) x + 3 y = 5$ and $2 x - 4 y + 1 = 0$ are perpendicular. Find the value of the constant $k$ .
[3 marks]
Answer $k = \dots\dots$
C3
The points $A (1, - 2)$ , $B (7, 1)$ , $C (5, 5)$ and $D (- 1, 2)$ are the vertices of a quadrilateral. Show that $A B C D$ is a rectangle, and find its area.
[5 marks]

Exam-style questions

Edexcel exam style

D1
Tom is asked for the line through $(5, 1)$ perpendicular to $l : 2 x - 3 y + 6 = 0$ . He writes: the gradient of $l$ is $\frac{2}{3}$ , so the perpendicular gradient is $- \frac{2}{3}$ ; $y - 1 = - \frac{2}{3} (x - 5)$ , giving $2 x + 3 y - 13 = 0$ .
Tom is wrong.
(a)
Show that Tom's line is not perpendicular to $l$ , and explain his mistake.
[2 marks]
(b)
Find a correct equation of the line in the form $a x + b y + c = 0$ , where $a$ , $b$ and $c$ are integers.
[3 marks]
Answer
(c)
Hence find the shortest distance from the point $(5, 1)$ to the line $l$ . Give your answer as an exact surd.
[4 marks]
Answer
Total for D1: 9 marks

D2
The line $l_{1}$ has equation $x + 2 y - 10 = 0$ and crosses the $x$ -axis at $B$ . The line $l_{2}$ passes through the origin $O$ and is perpendicular to $l_{1}$ . The lines $l_{1}$ and $l_{2}$ meet at $C$ .
Not drawn accurately
(a)
Find an equation of $l_{2}$ .
[2 marks]
Answer $y = \dots\dots$
(b)
Find the coordinates of $C$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
(c)
Find the lengths $O C$ and $C B$ , giving each in the form $k 5$ .
[3 marks]
Answer
(d)
Hence find the area of triangle $O C B$ .
[2 marks]
Answer
Total for D2: 9 marks
D3
The point $A$ has coordinates $(1, 5)$ . The points $B$ and $C$ lie on the line $y = x - 2$ , and each is a distance $26$ from $A$ . The $x$ -coordinate of $B$ is less than the $x$ -coordinate of $C$ .
(a)
Show that the $x$ -coordinates of $B$ and $C$ satisfy $x^{2} - 8 x + 12 = 0$ .
[3 marks]
(b)
Hence find the coordinates of $B$ and $C$ .
[2 marks]
Answer $B (\dots\dots, \dots\dots)$ $C (\dots\dots, \dots\dots)$
(c)
Find the exact area of triangle $A B C$ .
[4 marks]
Answer
Total for D3: 9 marks
D4
The lines $l_{1}$ and $l_{2}$ have equations $3 x + 4 y - 7 = 0$ and $6 x + 8 y + 36 = 0$ . Show that $l_{1}$ and $l_{2}$ are parallel, and find the shortest distance between them.
[5 marks]
Answer

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The points $A (- 1, 2)$ and $C (5, 10)$ are opposite corners of a square $A B C D$ . Find $B$ , $D$ and the area of the square.
[6 marks]
Hint 1 · What to try
The diagonals cross at right angles.
Hint 2 · The first line
$C$ is $3$ across, $4$ up from $M (2, 6)$ .
Hint 3 · The full method
$B$ and $D$ are $M \pm (- 4, 3)$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can test for parallel lines.
I can find a perpendicular line.
I can find exact distances.
I can find areas on a grid.

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