Similar shapes (lengths) Quiz (Grade 5)

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Score: 0 / 10

Similar shapes (lengths) — Notes & Worked Examples Show / Hide

Explanation:

Two shapes are mathematically similar if one is an enlargement of the other: corresponding angles are equal and corresponding sides are in the same ratio. First find the scale factor k = new lengthold length using two corresponding known sides.

Multiply by the scale factor to find a missing side on the larger shape, or divide by the scale factor to find a side on the smaller shape. When a smaller triangle is inside a larger triangle with parallel bases, sketch the two triangles separately using the full side length of the large triangle.

Example 1: Rectangles A and B are similar. Width of A is 6 cm and width of B is 15 cm. Length of B is 22.5 cm. Find the length of A.

Solution:

Scale factor k = 15 ÷ 6 = 2.5

Divide for smaller shape A:

Length A = 22.5 ÷ 2.5 = 9 cm

Example 2: In ▵ABC, DE is parallel to BC (D on AB, E on AC). Given AD = 4 cm, DB = 6 cm and DE = 5 cm, find BC.

Solution:

Full AB = 4 + 6 = 10 cm

k = 10 ÷ 4 = 2.5

BC = 5 × 2.5 = 12.5 cm

One Page Guide (PDF) Similar shapes (lengths) Worksheets

Question 1 AQA Nov 2020 3F Q26 / 3H Q9

[2 marks]

These two triangles are similar.

Work out the value of a.

Answer cm

Worked Solution & Answer

Corresponding sides between angles y and x are 20 cm and 16 cm M1
Linear scale factor = 1620 = 0.8
Corresponding side a = 12 × 0.8 = 9.6 cm. A1

Answer: Answer 9.6 cm

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Question 2

[2 marks]

Right-angled triangles with sides (7.5 cm, 10 cm, 12.5 cm) and (3 cm, 4 cm, 5 cm). Show that they are mathematically similar.

Show that these two triangles are mathematically similar.

Worked Solution & Answer

7.5 / 3 = 2.5, 10 / 4 = 2.5, 12.5 / 5 = 2.5. Since all three pairs of corresponding sides share the same scale factor of 2.5, the triangles are similar. M1 C1

Answer: 7.5 ÷ 3 = 2.5, 10 ÷ 4 = 2.5, and 12.5 ÷ 5 = 2.5; all corresponding sides are in the same ratio (scale factor 2.5)

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Question 3

[3 marks]

ABC and DAB are similar isosceles triangles.

AB = AC
AD = BD
BC : CD = 4 : 21
Find the ratio AB : AD

Worked Solution & Answer

Let BC = 4k and CD = 21k so BD = 4k + 21k = 25k M1
Since AD = BD, AD = 25k M1
By similar triangles ABC and DAB: AB / BC = AD / AB ⇒ AB² = BC × BD = 4k × 25k = 100k² ⇒ AB = 10k
Ratio AB : AD = 10k : 25k = 2 : 5. A1

Answer: 2 : 5

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Question 4 Specimen Set 2 1H Q18

[4 marks]

LMN is a right-angled triangle.

Angle NLM = 90°
PQ is parallel to LM.
The area of triangle PNQ is 8 cm²
The area of triangle LPQ is 16 cm²
Work out the area of triangle LQM.

cm²

Worked Solution & Answer

(The area of a triangle is ½ × base × height)
Triangles PNQ and LPQ share height PQ, so LP : PN = 16 : 8 = 2 : 1. Thus NL : NP = 3 : 1. Since ΔNLM is similar to ΔNPQ with length scale factor 3, Area(ΔNLM) = 8 × 3² = 72 cm² M3
Therefore Area(ΔLQM) = 72 – 8 – 16 = 48 cm². A1

Answer: 48 cm²

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Question 5 IGCSE Nov 2021 2F Q20 / 2H Q4

[4 marks]

ABC and DEF are similar triangles.

(a) Work out the length of AB. [2 marks]

(b) work out the length of EF. [2 marks]

Worked Solution & Answer

(a)

Scale factor = 45 ÷ 20 = 2.25 M1
⇒ AB = 36 × 2.25 = 81 cm A1

Answer (a): 81 cm

(b)

EF = 54 ÷ 2.25 = 24 cm M1 A1

Answer (b): 24 cm

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Question 6

[4 marks]

ABC and EDC are straight lines. EA is parallel to DB. EC = 8.1 cm, DC = 5.4 cm, DB = 2.6 cm. (a) Work out the length of AE. (2) AC = 6.15 cm. (b) Work out the length of AB. (2)

ABC and EDC are straight lines.
EA is parallel to DB.
EC = 8.1 cm.
DC = 5.4 cm.
DB = 2.6 cm.

(a) Work out the length of AE. [2 marks]

(b) Work out the length of AB. [2 marks]

Worked Solution & Answer

(a)

Scale factor = 8.1 ÷ 5.4 = 1.5. AE = 2.6 × 1.5 = 3.9 cm M1 A1

Answer (a): 3.9 cm

(b)

BC = 6.15 ÷ 1.5 = 4.1 cm M1
AB = 6.15 – 4.1 = 2.05 cm A1

Answer (b): 2.05 cm

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Question 7

[4 marks]

Triangles ACE and BCD are similar. EC = 8.1 cm, DC = 5.4 cm, DB = 2.6 cm. (a) Work out AE. (2) (b) AC = 6.15 cm. Work out AB. (2)

ABC and EDC are straight lines.
EA is parallel to DB.
EC = 8.1 cm.
DC = 5.4 cm.
DB = 2.6 cm.

(a) Work out the length of AE. [2 marks]

(b) Work out the length of AB. [2 marks]

Worked Solution & Answer

(a)

Scale factor = 8.1 ÷ 5.4 = 1.5 M1
AE = 2.6 × 1.5 = 3.9 cm A1

Answer (a): 3.9 cm

(b)

BC = 6.15 ÷ 1.5 = 4.1 cm M1
AB = 6.15 – 4.1 = 2.05 cm A1

Answer (b): 2.05 cm

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Question 8

[4 marks]

Triangle ABC and triangle DEF are similar.

(a) Work out the length of EF. [2 marks]

(b) Work out the length of AB. [2 marks]

Worked Solution & Answer

(a)

EF = 4 × 4 = 16 cm M1 A1

Answer (a): 16 cm

(b)

AB = 22 ÷ 4 = 5.5 cm M1 A1

Answer (b): 5.5 cm

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Question 9

[4 marks]

Triangle ABC and triangle DEF are similar.

(a) Work out the length of DF. [2 marks]

(b) Work out the length of CB. [2 marks]

Worked Solution & Answer

(a)

DF = 6.4 × 1.5 = 9.6 cm M1 A1

Answer (a): 9.6 cm

(b)

CB = 15 ÷ 1.5 = 10 cm M1 A1

Answer (b): 10 cm

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Question 10

[5 marks]

The two triangles in the diagram are similar.

There are two possible values of x.
Work out each of these values.
State any assumptions you make in your working.

Worked Solution & Answer

Case 1 (assuming BE is parallel to CD / ∠ABE = ∠ACD): (8 + x)/8 = 15/12 M1
⇒ 8 + x = 10 M1
⇒ x = 2. Case 2 (assuming ∠ABE = ∠ADC and ∠AEB = ∠ACD): (8 + x)/12 = 15/8 M1
⇒ 8 + x = 22.5 M1
⇒ x = 14.5. A1

Answer: x = 2 (assuming BE is parallel to CD, so ∠ ABE = ∠ ACD) and x = 14.5 (assuming ∠ ABE = ∠ ADC)

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Score: 0 / 10 (0 / 36 marks)

Similar shapes (lengths) Worksheets Next: Trigonometry (SOHCAHTOA) →

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5S - Spheres and cones	5T - Similar shapes (lengths)
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