State whether the following pair of triangles are congruent: Triangle PQR and triangle XYZ, where P matches X, Q matches Y, R matches Z, and all three corresponding sides are equal in length.
(1)
2
Give the congruence condition name for the following: two triangles with two angles equal and the included side equal.
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3
Triangle A and triangle B are shown with angles 50 degrees and 50 degrees, and a side between them equal. Are the triangles congruent? State the reason.
(2)
4
Write down whether the pair of shapes are similar or not: two rectangles, one 3 cm by 6 cm and the other 6 cm by 12 cm.
(1)
5
A triangle has sides 5 cm, 7 cm and 9 cm. A second triangle has sides 10 cm, 14 cm and 18 cm. Are the triangles similar? Explain briefly.
(2)
6
A small model of a bridge is similar to the real bridge with scale 1:150. On the model the arch height is 4 cm. Find the actual arch height in metres.
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7
Two similar triangles have corresponding sides 8 cm and 12 cm. The small triangle has another side of length 10 cm. Find the matching side length in the larger triangle.
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8
Triangle ABC is congruent to triangle DEF. Point B corresponds to E. If AB = 6 cm and EF = 6 cm, give the length of DE if AC = 10 cm and DF corresponds to AC.
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9
A triangle has sides 6 cm, 8 cm and 10 cm. A similar triangle has shortest side 9 cm. Find the other two sides of the similar triangle.
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10
Rectangle R is similar to rectangle S. R measures 4 cm by 9 cm. S has a width of 6 cm. Find the length of S.
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11
Two triangles are similar. The ratio of areas is 9:16. Find the scale factor between lengths of the triangles.
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12
A rectangle measures 7 cm by 15 cm. A similar rectangle has its shorter side 5.6 cm. Find the longer side of the similar rectangle. Give your answer to 1 decimal place.
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13
Triangles PQR and STU have PQ = 5 cm, QR = 7 cm, PR = 8 cm and ST = 5 cm, TU = 7 cm, SU = 8 cm. Are the triangles congruent or just similar? Give a brief reason.
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14
Two similar polygons have perimeters in the ratio 5:8. Find the ratio of their corresponding side lengths and the ratio of their areas.
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15
A map uses a scale where 1 cm represents 2 km. Two similar triangular fields on the map have corresponding heights 3.5 cm and 5.25 cm. Find the actual difference in height of the fields in metres.
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16
Triangle ABC has angles 30 degrees, 60 degrees and 90 degrees. Triangle DEF also has angles 30 degrees, 60 degrees and 90 degrees. Are the triangles similar? Explain briefly, including what this tells you about the side lengths.
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17
Triangles XYZ and PQR are similar. In triangle XYZ, XY = 6 cm, YZ = 9 cm. In triangle PQR, PQ = 8 cm and QR corresponds to YZ. Find PR in triangle PQR given that PR corresponds to XZ which is 12 cm in triangle XYZ.
(4)
18
Stretch question. A garden planter is an enlargement of a smaller template by a scale factor of 7/4. The template has area 32 cm2. Find the area of the planter. Then, if the planter has a side that measures 17.5 cm, find the matching side length on the template. Show your working.
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Mark scheme · KS3.M-G18 Congruence and Similarity
Question 1
B1 Correct statement: yes, they are congruent (SSS) cao
Answer: Yes, they are congruent (SSS).
Question 2
B1 Correct name: ASA cao
Answer: ASA
Question 3
M1 Identify that two angles and the included side are equal or equivalent reasoning
A1 Conclusion: triangles are congruent by ASA cao
Answer: Yes, congruent by ASA.
Question 4
B1 Correct statement: similar, sides are in proportion 1:2 cao
Answer: Similar (scale factor 2).
Question 5
M1 Shows side ratios are equal: 10/5 = 14/7 = 18/9 = 2 or equivalent
A1 States they are similar with scale factor 2 cao
Answer: Yes. All corresponding sides have ratio 2, so similar with scale factor 2.
Question 6
M1 Method: multiply 4 cm by 150 to get real height in cm or state 4 x 150 = 600 cm
A1 Correct conversion to metres: 6 m cao
B1 Units stated: metres cao
Answer: 6 m
Question 7
M1 Method: state scale factor = 12/8 = 3/2 or 1.5
M1 Multiply 10 by scale factor to get 15 (allow equivalent method)
A1 Answer 15 cm cao
Answer: 15 cm
Question 8
M1 Use congruence: corresponding sides equal, so DE corresponds to AB or AC as stated
A1 Correct length: DE = 6 cm cao
Answer: 6 cm
Question 9
M1 Find scale factor: 9/6 = 3/2
M1 Multiply 8 and 10 by 3/2 to get 12 and 15
A1 Correct pair: 12 cm and 15 cm cao
Answer: 12 cm and 15 cm
Question 10
M1 Find scale factor: 6/4 = 3/2
M1 Multiply 9 by 3/2 to get 13.5
A1 Answer 13.5 cm cao
Answer: 13.5 cm
Question 11
M1 Recognise that area ratio = (scale factor)2 and take square roots: √9/16 = 3/4
A1 State scale factor 3:4 or 3/4 cao
Answer: 3/4
Question 12
M1 Find scale factor: 5.6/7 = 0.8
M1 Multiply 15 by 0.8 to get 12
A1 Answer 12.0 cm or 12.0 cao
Answer: 12.0 cm
Question 13
M1 Recognise that all three corresponding sides are equal
A1 State congruent by SSS cao
Answer: Congruent, because all three corresponding sides are equal (SSS).
Question 14
M1 Recognise perimeter ratio equals linear scale factor: 5:8
M1 State scale factor 5/8 or 5:8 for lengths
A1 Area ratio = (5/8)2 = 25/64 cao
Answer: Lengths 5:8; areas 25:64
Question 15
M1 Find map difference: 5.25 - 3.5 = 1.75 cm
M1 Convert to real distance: 1.75 cm -> 1.75 x 2 km = 3.5 km
A1 Convert to metres: 3.5 km = 3500 m cao
Answer: 3500 m
Question 16
M1 Recognise that equal corresponding angles give similarity by AA
A1 State triangles are similar because corresponding angles are equal (AA), so corresponding side lengths must be in the same proportion; accept correct explanation
Answer: They are similar because their corresponding angles are equal (AA), so their corresponding side lengths must be in the same proportion (a fixed ratio).