KS3 Maths · Year 8 · Chapter 6: Pythagoras' theorem · Lesson 3

The converse of Pythagoras' theorem

Year 8About 70 minutesCalculator allowed45 marks

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Date

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Quick questions to start.

A1
Needed today
Work out $2. 5^{2} + 1. 2^{2}$ .
[1 mark]
Answer
A2
Needed today
Write down the value of $225$ .
[1 mark]
Answer
A3
Last lesson
A $6$ m ladder stands $2$ m from a wall. How high does it reach, to $1$ d.p.?
[1 mark]
Answer
A4
Year 7
A triangle has angles $9 0^{\circ}$ and $3 7^{\circ}$ . Find the third.
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it. Call the longest side $c$ . If $a^{2} + b^{2} = c^{2}$ , the angle opposite $c$ is a right angle.

Example 1

Is this triangle right-angled?

Not drawn accurately

Hint Square the two shorter sides and add. Compare with the square of the longest side.

B1
Is this triangle right-angled? Show your working.
[3 marks]
Not drawn accurately

Example 2

Is the largest angle of this triangle acute, right or obtuse?

Not drawn accurately

Hint If

a^{2} + b^{2} < c^{2}

, the longest side is too long for a right angle, so the angle opposite it is obtuse.

B2
Is the largest angle of this triangle acute, right or obtuse? Show your working.
[3 marks]
Not drawn accurately

Example 3

A gate frame is $1.2$ m wide and $1.6$ m high. Its diagonal measures $2.05$ m. Is the frame a rectangle?

Not drawn accurately

Hint A rectangle has right angles at its corners. Test one corner with the converse.

B3
A door frame is $0.72$ m wide and $2.1$ m high. Its diagonal measures $2.22$ m. Is the frame a rectangle? Show your working.
[3 marks]
Not drawn accurately

Example 4

Angle $A B C = 9 0^{\circ}$ . Show that angle $A C D = 9 0^{\circ}$ .

Not drawn accurately

Hint First use Pythagoras in triangle

A B C

to find

A C

. Then test triangle

A C D

B4
Angle $A B C = 9 0^{\circ}$ . Show that angle $A C D = 9 0^{\circ}$ .
[3 marks]
Not drawn accurately

Practice

Year 8

C1
In triangle $P QR$ , $P Q = 11$ cm, $QR = 60$ cm and $P R = 61$ cm. Show that the triangle is right-angled, and say which angle is the right angle.
[2 marks]
Not drawn accurately
C2
These are the sides of four triangles. Which one is NOT right-angled? Show how you know. (a) $9, 40, 41$ (b) $15, 36, 39$ (c) $10, 22, 25$ (d) $16, 30, 34$
[2 marks]

C3
Jo walks $3.6$ km due north, turns, then walks $1.5$ km. She is now $3.9$ km from her start. Did she turn through a right angle? Show your working.
[3 marks]
Not drawn accurately
C4
A triangle has sides $1.6$ m, $3$ m and $3.4$ m. Show that it is right-angled. Then work out its area.
[3 marks]
Not drawn accurately

Exam-style questions

GCSE Foundation

D1
Ben draws a quadrilateral with all four sides $9$ cm. The diagonal $A C$ is $12.6$ cm. Ben says, "My quadrilateral is a square."
Is Ben right? You must show your working.
[3 marks]

D2
$P$ is the point $(3, 2)$ , $Q$ is the point $(6, 8)$ and $R$ is the point $(12, 5)$ .
(a)
Work out $P Q^{2}$ , $Q R^{2}$ and $P R^{2}$ .
[2 marks]
(b)
Show that triangle $P QR$ is right-angled. Which angle is the right angle?
[2 marks]
Total for D2: 4 marks
D3
$A$ , $B$ and $D$ lie on a straight line. $C B$ is perpendicular to $A D$ . $A B = 18$ cm, $B D = 8$ cm and $C B = 12$ cm.
(a)
Work out $A C^{2}$ and $C D^{2}$ .
[2 marks]
(b)
Show that angle $A C D = 9 0^{\circ}$ .
[2 marks]
(c)
Work out the area of triangle $A C D$ .
[1 mark]
Answer $\dots$ cm²
Total for D3: 5 marks
D4
A builder checks the corner of a room. She marks a point $0.9$ m along one wall and a point $1.2$ m along the other wall. The two points are $1.53$ m apart. Is the corner a right angle? If not, is it more or less than $9 0^{\circ}$ ? Show your working.
[3 marks]

Extension

Stretch

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

E1
A triangle has sides $9$ cm, $12$ cm and $x$ cm, where $x$ is a whole number. For how many values of $x$ is the triangle obtuse?
[4 marks]
Hint 1 · What to try
The third side must be more than $12 - 9$ and less than $12 + 9$ . So $x$ is one of $4, 5, \dots, 20$ .
Hint 2 · The first line
If $x$ is the longest side, the triangle is obtuse when $x^{2} > 9^{2} + 1 2^{2} = 225$ . If $12$ is the longest side, it is obtuse when $9^{2} + x^{2} < 144$ .
Hint 3 · The full method
$x^{2} > 225$ gives $x = 16, 17, 18, 19, 20$ . $x^{2} < 63$ gives $x = 4, 5, 6, 7$ . That is $9$ values.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the converse of Pythagoras' theorem to test for a right angle.
I can tell whether the largest angle is acute, right or obtuse.
I can check a frame, a walk or a corner for a right angle.
I can use Pythagoras' theorem and its converse in one problem.

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