OCR GCSE Maths · OCR 9

Congruence and Similarity

50 maths questions with worked answers with answers for OCR 9 Congruence and Similarity in OCR GCSE Maths (spec J560). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official OCR material.

All 50 questions and answers

Question 1 · Reflections · OCR 9.01a
Reflect the point (3, 5) in the y-axis.
Answer:
The x-coordinate changes sign.
(−3, 5).
Question 2 · Reflections · OCR 9.01a
Reflect the point (2, −4) in the line y = x.
Answer:
Swap the coordinates.
(−4, 2).
Question 3 · Rotations · OCR 9.01b
Rotate (4, 1) by 90° clockwise about the origin.
Answer:
(x, y) becomes (y, −x).
(1, −4).
Question 4 · Rotations · OCR 9.01b
Rotate (3, 2) by 180° about the origin.
Answer:
Both coordinates change sign.
(−3, −2).
Question 5 · Describing transformations · OCR 9.01b
What three details must you give to describe a rotation?
Answer:
Centre, angle and direction.
For example: 90° anticlockwise about (0, 0).
Question 6 · Transformations · OCR 9.01b
Are two shapes necessarily congruent after a rotation?
Answer:
A rotation preserves every length and angle.
The image has the same shape and size, so it is congruent.
Question 7 · Translations · OCR 9.01c
Translate (5, −2) by −34.
Answer:
5 − 3 = 2 and −2 + 4 = 2.
(2, 2).
Question 8 · Column vectors · OCR 9.01c
What does the column vector 3−2 represent?
Answer:
A movement.
3 units right and 2 units down.
Question 9 · Column vectors · OCR 9.01c
Write the column vector for a movement of 4 left and 5 up.
Answer:
Left is negative x; up is positive y.
−45.
Question 10 · Vectors and points · OCR 9.01c
A is (2, 3) and B is (7, 1). Write the vector AB.
Answer:
Subtract the coordinates of A from B: 7 − 2 = 5 and 1 − 3 = −2.
AB = 5−2.
Question 11 · Vectors and points · OCR 9.01c
Point P is (1, −4). Translate it by −36.
Answer:
1 − 3 = −2 and −4 + 6 = 2.
(−2, 2).
Question 12 · Diagrams · OCR 9.01c
How is a vector shown on a diagram?
Answer:
A line segment with an arrow.
The arrow shows the direction; the length shows the magnitude.
Question 13 · Notation · OCR 9.01c
How is a vector written in print and by hand?
Answer:
In print it is bold: a.
By hand it is underlined: a̲. A vector from A to B is written AB with an arrow above.
Question 14 · Negative vectors · OCR 9.01c
If AB = a, what is BA?
Answer:
BA goes in the opposite direction.
BA = −a.
Question 15 · Combined transformations · OCR 9.01d · Higher only
What single transformation is a reflection in the x-axis followed by a reflection in the y-axis?
Answer:
(x, y) → (x, −y) → (−x, −y).
A rotation of 180° about the origin.
Question 16 · Invariant points · OCR 9.01d · Higher only
What is an invariant point?
Answer:
A point that does not move under a transformation.
For example, points on the mirror line in a reflection.
Question 17 · Congruence · OCR 9.02a
State the four conditions for triangles to be congruent.
Answer:
SSS, SAS, ASA (or AAS) and RHS.
RHS = right angle, hypotenuse, side.
Question 18 · Congruence · OCR 9.02a
Is AAA a valid condition for congruence? Explain.
Answer:
Three equal angles only show the triangles are the same shape.
No; the triangles are similar but may be different sizes.
Question 19 · Congruence · OCR 9.02a
What does congruent mean?
Answer:
Compare corresponding sides and angles.
Congruent shapes match exactly in shape and size; orientation may differ.
Question 20 · Congruence · OCR 9.02a
Two triangles have corresponding side lengths 3 cm, 4 cm and 5 cm. Are they congruent?
Answer:
The three pairs of corresponding sides have equal lengths.
Therefore the triangles are congruent by SSS.
Question 21 · Congruence · OCR 9.02b
Do three equal corresponding angles alone prove two triangles congruent?
Answer:
Equal angles establish the same shape but do not fix the side lengths.
The triangles can have different sizes, so this only proves similarity.
Question 22 · Scalar multiples · OCR 9.03a
a = 3−2. Find 4a.
Answer:
Multiply each component by 4.
12−8.
Question 23 · Combining · OCR 9.03a
p = 52 and q = 1−3. Find 3p − 2q.
Answer:
3p = 156 and 2q = 2−6.
1312.
Question 24 · Magnitude · OCR 9.03a · Higher only
Find the magnitude of 68.
Answer:
Use Pythagoras: √(6² + 8²) = √100.
10.
Question 25 · Vector paths · OCR 9.03a
OA = a and OB = b. Find AB.
Answer:
AB = AO + OB.
AB = −a + b = b − a.
Question 26 · Vector paths · OCR 9.03a
OA = a and OB = b. M is the midpoint of AB. Find AM.
Answer:
AB = b − a.
AM = 12(b − a).
Question 27 · Ratio on a line · OCR 9.03a · Higher only
OA = a and OB = b. P lies on AB with AP : PB = 1 : 2. Find OP.
Answer:
AP = 13AB = 13(b − a).
OP = a + 13(b − a) = 23a + 13b.
Question 28 · Collinear points · OCR 9.03a · Higher only
AB = 2a + 4b and BC = a + 2b. Show that A, B and C are collinear.
Answer:
AB = 2(a + 2b) = 2BC.
AB is parallel to BC and they share B, so A, B and C are collinear.
Question 29 · Parallelograms · OCR 9.03a · Higher only
In parallelogram OABC, OA = a and OC = c. Find OB.
Answer:
OB = OA + AB, and AB = OC = c.
OB = a + c.
Question 30 · Adding vectors · OCR 9.03b
a = 25 and b = 3−1. Find a + b.
Answer:
Add the top and bottom numbers separately.
54.
Question 31 · Parallel vectors · OCR 9.03b
Are 23 and 69 parallel? Explain.
Answer:
69 = 3 × 23.
Yes; one is a multiple of the other.
Question 32 · Parallel vectors · OCR 9.03b
Are 4−2 and −21 parallel?
Answer:
4−2 = −2 × −21.
Yes; they are parallel but point in opposite directions.
Question 33 · Parallel vectors · OCR 9.03b · Higher only
a = 3k is parallel to 610. Find k.
Answer:
610 = 2 × 35.
k = 5.
Question 34 · Collinear points · OCR 9.03b · Higher only
How do you prove three points A, B and C are collinear using vectors?
Answer:
Show that AB is a multiple of BC (or AC).
They share point B, so the points lie on a straight line.
Question 35 · Equal vectors · OCR 9.03b
When are two vectors equal?
Answer:
When they have the same magnitude and the same direction.
Their position does not matter.
Question 36 · Solving with vectors · OCR 9.03b
x3 + 4y = 9−1. Find x and y.
Answer:
x + 4 = 9 and 3 + y = −1.
x = 5, y = −4.
Question 37 · Similar triangles · OCR 9.04a
Triangles ABC and DEF are similar. AB = 5 cm, DE = 15 cm and BC = 7 cm. Find EF.
Answer:
Scale factor = 15 ÷ 5 = 3.
EF = 7 × 3 = 21 cm.
Question 38 · Transformations · OCR 9.04a
Are two shapes similar after enlargement by scale factor 2?
Answer:
An enlargement by 2 doubles every length and preserves every angle.
Corresponding sides are in a constant ratio, so the shapes are similar.
Question 39 · Similarity · OCR 9.04a
Similar triangles have corresponding side lengths 3 cm and 9 cm. Find the scale factor from the smaller to the larger.
Answer:
Scale factor = corresponding larger length ÷ smaller length.
9 ÷ 3 = 3.
Question 40 · Similarity · OCR 9.04a
Similar figures have side lengths in ratio 2:5. A small side is 6 cm. Find its corresponding large side.
Answer:
Scale factor from small to large = 52.
6 × 52 = 15 cm.
Question 41 · Enlargements · OCR 9.04b
Enlarge the point (2, 3) by scale factor 3 with centre the origin.
Answer:
Multiply each coordinate by 3.
(6, 9).
Question 42 · Enlargements · OCR 9.04b · Higher only
Enlarge the point (4, 6) by scale factor −12 about the origin.
Answer:
Multiply each coordinate by −0.5.
(−2, −3).
Question 43 · Enlargements · OCR 9.04b
What does a fractional scale factor between 0 and 1 do?
Answer:
It makes the shape smaller.
The image is closer to the centre of enlargement.
Question 44 · Similar shapes · OCR 9.04c
Two similar rectangles have widths 4 cm and 10 cm. The smaller has length 6 cm. Find the larger length.
Answer:
Scale factor = 104 = 2.5.
6 × 2.5 = 15 cm.
Question 45 · Similar shapes · OCR 9.04c · Higher only
Two similar shapes have length scale factor 3. What is the area scale factor?
Answer:
Area scale factor = (length scale factor)².
3² = 9.
Question 46 · Similar shapes · OCR 9.04c · Higher only
Two similar solids have volumes 8 cm³ and 125 cm³. Find the length scale factor.
Answer:
Length scale factor = ∛(volume scale factor) = ∛1258.
52 = 2.5.
Question 47 · Similar shapes · OCR 9.04c · Higher only
Two similar jugs have heights 10 cm and 15 cm. The smaller holds 400 ml. How much does the larger hold?
Answer:
Length scale factor 1.5, so volume scale factor 1.5³ = 3.375.
400 × 3.375 = 1350 ml.
Question 48 · Similarity · OCR 9.04c
What is the difference between congruent and similar shapes?
Answer:
Congruent shapes are identical in shape and size.
Similar shapes have the same shape but are different sizes, with sides in proportion.
Question 49 · Similarity · OCR 9.04c
Triangle A has sides 3, 4, 5. Triangle B has sides 6, 8, 10. Are they similar? Explain.
Answer:
Each side of B is 2 × the matching side of A.
Yes; the sides are in the same ratio.
Question 50 · Similarity · OCR 9.04c
Two similar shapes have length scale factor 4. A side is 7 cm on the smaller shape. Find its corresponding larger side.
Answer:
Multiply the smaller length by 4.
7 × 4 = 28 cm.

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