State whether the following pair of triangles are congruent: Triangle DEF and triangle UVW, where D matches U, E matches V, F matches W, and all three corresponding sides are equal in length.
(1)
2
Give the congruence condition name for the following: two triangles with two sides equal and the included angle equal.
(1)
3
Triangle C and triangle D are shown with angles 65 degrees and 65 degrees, and the 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 4 cm by 8 cm and the other 8 cm by 16 cm.
(1)
5
A triangle has sides 6 cm, 9 cm and 12 cm. A second triangle has sides 12 cm, 18 cm and 24 cm. Are the triangles similar? Explain briefly.
(2)
6
A small model of a building is similar to the real building with scale 1:200. On the model the tower height is 5 cm. Find the actual tower height in metres.
(2)
7
Two similar triangles have corresponding sides 6 cm and 15 cm. The small triangle has another side of length 8 cm. Find the matching side length in the larger triangle.
(2)
8
Triangle ABC is congruent to triangle DEF. Point B corresponds to E. If AB = 7 cm and EF = 7 cm, give the length of DE if AC = 11 cm and DF corresponds to AC.
(2)
9
A triangle has sides 8 cm, 10 cm and 14 cm. A similar triangle has shortest side 12 cm. Find the other two sides of the similar triangle.
(2)
10
Rectangle R is similar to rectangle S. R measures 5 cm by 11 cm. S has a width of 8 cm. Find the length of S, to 1 decimal place.
(2)
11
Give the congruence condition name for the following: a right-angled triangle where the hypotenuse and one other side match another right-angled triangle.
(1)
12
A rectangle measures 4 cm by 10 cm. A second rectangle measures 6 cm by 15 cm. Are the two rectangles similar? Explain briefly, stating the scale factor.
(2)
13
A rectangle measures 9 cm by 20 cm. A similar rectangle has its shorter side 6.3 cm. Find the longer side of the similar rectangle. Give your answer to 1 decimal place.
(3)
14
Two similar triangles have a ratio of areas of 25:36. The smaller triangle has a side of length 10 cm.
(a)Find the scale factor between the lengths of the two triangles.(2)
(b)Hence find the corresponding side of the larger triangle.(1)
15
Two similar polygons have perimeters in the ratio 6:11. Find the ratio of their corresponding side lengths and the ratio of their areas.
(3)
16
A map uses a scale where 1 cm represents 2.5 km. Two similar triangular fields on the map have corresponding heights 3.6 cm and 5.2 cm. Find the actual difference in height of the fields in metres.
(3)
17
Triangles XYZ and PQR are similar. In triangle XYZ, XY = 8 cm, YZ = 11 cm. In triangle PQR, PQ = 10 cm and QR corresponds to YZ. Find PR in triangle PQR given that PR corresponds to XZ which is 14 cm in triangle XYZ.
(4)
18
Stretch question. A garden planter is an enlargement of a smaller template by a scale factor of 5/3. The template has area 45 cm2. Find the area of the planter. Then, if the planter has a side that measures 20 cm, find the matching side length on the template. Show your working.
(4)
Mark scheme · KS3.M-G18D Congruence and Similarity: Fluency and Exam Drill
Question 1
B1 Correct statement: yes, they are congruent (SSS) cao
Answer: Yes, they are congruent (SSS).
Question 2
B1 Correct name: SAS cao
Answer: SAS
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: 12/6 = 18/9 = 24/12 = 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 5 cm by 200 to get real height in cm, 5 x 200 = 1000 cm
A1 Correct conversion to metres: 10 m cao
Answer: 10 m
Question 7
M1 Method: state scale factor = 15/6 = 2.5
A1 Answer 20 cm cao
Answer: 20 cm
Question 8
M1 Use congruence: corresponding sides equal, so DE corresponds to AB
A1 Correct length: DE = 7 cm cao
Answer: 7 cm
Question 9
M1 Find scale factor: 12/8 = 3/2
A1 Correct pair: 15 cm and 21 cm cao
Answer: 15 cm and 21 cm
Question 10
M1 Find scale factor: 8/5 = 1.6
A1 Answer 17.6 cm cao
Answer: 17.6 cm
Question 11
B1 Correct name: RHS cao
Answer: RHS
Question 12
M1 Shows both pairs of sides in the same ratio: 6/4 = 1.5 and 15/10 = 1.5
A1 States similar with scale factor 1.5 cao
Answer: Yes, similar with scale factor 1.5.
Question 13
M1 Find scale factor: 6.3/9 = 0.7
M1 Multiply 20 by 0.7 to get 14
A1 Answer 14.0 cm or 14.0 cao
Answer: 14.0 cm
Question 14
(a) M1 Recognise that area ratio = (scale factor)2 and take square roots: √25/36 = 5/6
(a) A1 State scale factor 5:6 or 5/6 cao
(a) Answer: 5/6
(b) B1 12 cm cao ft, using 10 x 6/5
(b) Answer: 12 cm
Question 15
M1 Recognise perimeter ratio equals linear scale factor: 6:11
M1 State scale factor 6/11 or 6:11 for lengths
A1 Area ratio = (6/11)2 = 36/121 cao
Answer: Lengths 6:11; areas 36:121
Question 16
M1 Find map difference: 5.2 - 3.6 = 1.6 cm
M1 Convert to real distance: 1.6 cm -> 1.6 x 2.5 km = 4 km