Year 9 Shape and Geometry Worksheet - Intermediate

Year 9 Shape and Geometry worksheet for Year 9 (Key Stage 3 / Year 7-9 (Ages 11-14)). Intermediate level maths practice, aligned to the UK National Curriculum. Print-ready with answer key included.

Maths•Key Stage 3 / Year 7-9 (Ages 11-14)•Geometryintermediate

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Year 9 Shape and Geometry: Right Triangles and Transformations

Ages 13–14 | Key Stage 3 | Intermediate

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Instructions: Show your working clearly and include units where appropriate. Give trigonometric answers to 1 decimal place unless the question asks for an exact value. For transformations, state the defining information precisely; for scale factors, explain how corresponding lengths change.

Pythagoras’ Theorem

Use a² + b² = c², where c is the hypotenuse. Show your substitution and calculation, then give each answer to 1 decimal place where necessary.

1. A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse. [3 marks]

2. A right-angled triangle has hypotenuse 13 m and one shorter side 5 m. Find the other shorter side. [3 marks]

3. A 10 m ladder rests against a vertical wall. Its foot is 6 m from the wall. How high up the wall does the ladder reach? [3 marks]

4. A rectangle is 9 cm long and 12 cm wide. Find the length of its diagonal. [3 marks]

Trigonometric Ratios

For each right-angled triangle, identify the opposite, adjacent and hypotenuse sides relative to the stated angle. Choose the appropriate ratio: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, or tan θ = opposite/adjacent. Give lengths to 1 decimal place and angles to 1 decimal place.

1. In a right-angled triangle, θ = 35° and the hypotenuse is 12 cm. Find the side opposite θ. [3 marks]

2. In a right-angled triangle, θ = 48° and the hypotenuse is 10 m. Find the side adjacent to θ. [3 marks]

3. In a right-angled triangle, θ = 27° and the side adjacent to θ is 8 cm. Find the side opposite θ. [3 marks]

4. In a right-angled triangle, the side opposite θ is 7 cm and the hypotenuse is 12 cm. Find θ. [3 marks]

Describe Transformations

State the transformation using the information given. For a translation, use a column vector; for a reflection, name the mirror line; for a rotation, give the angle, direction and centre. For an enlargement, give the scale factor and centre.

1. The point A(−2, 3) maps to A′(4, 1). Describe the translation using a column vector. [2 marks]

2. The point P(5, −3) maps to P′(−5, −3). Describe the reflection. [2 marks]

3. The point Q(3, 1) maps to Q′(−1, 3) under a rotation about the origin. Describe the rotation fully. [3 marks]

4. A shape is enlarged from the centre (0, 0) by scale factor 2.5. Describe the change in the lengths of its sides and its area. [3 marks]

Carry Out Transformations on Coordinates

Write your answer in the box below.

Similarity, Congruence and Scale Factors

Use corresponding sides to decide whether shapes are similar or congruent. For similar shapes, find the scale factor and use it to calculate missing lengths. Remember that congruent shapes have the same shape and size.

1. Triangle A has side lengths 4 cm, 6 cm and 8 cm. Triangle B has side lengths 6 cm, 9 cm and 12 cm. Are the triangles similar, congruent, or neither? Give the scale factor from A to B. [3 marks]

2. Two rectangles have dimensions 5 cm by 8 cm and 10 cm by 16 cm. Are they similar, congruent, or neither? Give the scale factor from the smaller rectangle to the larger. [3 marks]

3. Two similar triangles have corresponding sides 7 cm and 10.5 cm. The side corresponding to 7 cm on the first triangle is 12 cm on a second pair of similar triangles. Find the side corresponding to 10.5 cm. [3 marks]

4. A shape is enlarged by scale factor 3. A corresponding side on the image is 21 cm. Find the length of that side on the original shape. [2 marks]

Answer Key

Activity 1: 10 cm ; 12 m ; 8 m ; 15 cm

Activity 2: 6.9 cm ; 6.7 m ; 4.1 cm ; 35.7°

Activity 3: Translation by vector (6, −2) ; reflection in the y-axis ; 90° anticlockwise about the origin ; side lengths multiply by 2.5 and area multiplies by 6.25

Activity 4: A′(−1, 4), B′(2, 4), C′(0, 6) ; D′(3, 1), E′(1, 4), F′(2, 2) ; G′(−1, 2), H′(−1, 4), J′(−3, 3) ; K′(2, 2), L′(6, 2), M′(2, 4)

Activity 5: Similar; scale factor 1.5 ; similar; scale factor 2 ; 18 cm ; 7 cm