Unit notes and practice questions

Grade 9 Mathematics unit 7: Congruency and Similarity

Everything here is for unit 7 of the Grade 9 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.

4
Cards: 10
Practice questions on unit 7: 209–246
The textbook pages for unit 7

What unit 7 covers

1.Revision on Congruency of Triangles
2.Definition of Similar Figures
3.Theorems on Similar Plane Figures
4.Ratio of Perimeters of Similar Plane Figures
5.Ratio of Areas of Similar Plane Figures
6.Construction of Similar Plane Figures
7.Applications on Similarities

Short notes for unit 7

The cards filed under this chapter on the Grade 9 Mathematics page. Open one at a time.

Linear Scale Factor vs Area ScalingCommon mistake

When you use it

Check this whenever a word problem asks how changing side lengths changes the area or perimeter.

Watch out

Multiplying side lengths by 2 doubles the perimeter, but multiplies the area by 2^2 = 4.

Drafted from Grade 9 Mathematics, pages 209-246, then checked twice before it went up

Ratio of Areas of Similar FiguresFormula

\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2 = k^2

\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2 = k^2

$A_1$: Area of first similar figure $\text{m}^2$
$A_2$: Area of second similar figure $\text{m}^2$
$s_1$: side length of first figure $\text{m}$
$s_2$: corresponding side length of second figure $\text{m}$
$k$: scale factor $1$

When you use it

Use when finding the area of a similar plane figure given side lengths or the scale factor.

Watch out

Remember to square the scale factor. The ratio of areas is k^2 and not k.

Drafted from Grade 9 Mathematics, pages 209-246, then checked twice before it went up

Ratio of Perimeters of Similar FiguresFormula

\frac{P_1}{P_2} = \frac{s_1}{s_2} = k

\frac{P_1}{P_2} = \frac{s_1}{s_2} = k

$P_1$: Perimeter of first similar figure $\text{m}$
$P_2$: Perimeter of second similar figure $\text{m}$
$s_1$: side length of first figure $\text{m}$
$s_2$: corresponding side length of second figure $\text{m}$
$k$: scale factor $1$

When you use it

Use to find an unknown perimeter or side length when two geometric figures are similar.

Watch out

Match strictly corresponding sides in order when setting up the ratio.

Drafted from Grade 9 Mathematics, pages 209-246, then checked twice before it went up

Triangle Similarity CriteriaReference tableAA · SSS · SAS

Criterion	Given Equality or Ratio	Conclusion
AA	$\angle A = \angle D \text{ and } \angle B = \angle E$	$\triangle ABC \sim \triangle DEF$
SSS	$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$	$\triangle ABC \sim \triangle DEF$
SAS	$\frac{AB}{DE} = \frac{AC}{DF} \text{ and } \angle A = \angle D$	$\triangle ABC \sim \triangle DEF$

When you use it

Use these tests to prove two triangles are similar using minimum required angle or side conditions.

Watch out

In SAS similarity, the congruent angle must be the included angle between the two proportional sides.

Drafted from Grade 9 Mathematics, pages 209-246, then checked twice before it went up

Practice questions on unit 7

10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.

Question 1

According to the AA (Angle-Angle) similarity theorem, two triangles are similar if:

A.Two angles of one triangle are congruent to two angles of the other triangle
B.All three pairs of corresponding sides are equal in length
C.The two triangles have exactly equal perimeters
D.The two triangles have equal areas

Show the answer and why

Answer: A Two angles of one triangle are congruent to two angles of the other triangle

Why

The AA similarity theorem states that if two angles of one triangle are congruent to two angles of another triangle, the two triangles are similar.

Question 2

The ratio of the perimeters of two similar triangles is 3:5. What is the ratio of their corresponding sides?

A.9:25
B.3:5
C.6:15
D.1:2

Show the answer and why

Answer: B 3:5

Why

For any two similar plane figures, the ratio of their perimeters is equal to the ratio of their corresponding side lengths, which is 3:5.

Question 3

A giraffe is 6 m tall and casts a shadow of 4 m. At the same time, a smaller giraffe casts a shadow of 2 m. How tall is the smaller giraffe?

A.2 m
B.3 m
C.4 m
D.5 m

Show the answer and why

Answer: B 3 m

Why

Using the proportionality of similar triangles: Height / Shadow = 6 / 4 = x / 2. Solving for x gives x = (6 * 2) / 4 = 3 m.

Question 4

If the ratio of the side lengths of two similar rectangles is 3:1, what is the ratio of their areas?

A.3:1
B.6:1
C.9:1
D.12:1

Show the answer and why

Answer: C 9:1

Why

The ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding sides: 3² : 1² = 9:1.

Question 5

Given that ΔABC ~ ΔDEF, with AB = 9 cm, BC = 15 cm, and DE = 6 cm. What is the length of side EF?

A.8 cm
B.10 cm
C.12 cm
D.18 cm

Show the answer and why

Answer: B 10 cm

Why

Because the triangles are similar, the ratios of corresponding sides are equal: AB / DE = BC / EF, which means 9 / 6 = 15 / EF. Therefore, EF = (15 * 6) / 9 = 10 cm.

Question 6

Two similar triangles have corresponding sides of length 3 cm and 9 cm. If the area of the smaller triangle is 12 cm², what is the area of the larger triangle?

A.36 cm2
B.72 cm2
C.108 cm2
D.144 cm2

Show the answer and why

Answer: C 108 cm2

Why

The scale factor of side lengths is 9 / 3 = 3. The ratio of their areas is 3² = 9. Thus, the area of the larger triangle is 12 * 9 = 108 cm².

Question 7

In Hawassa, a flagpole casts a shadow 9 m long. At the same moment, a student who is 1.8 m tall standing nearby casts a shadow 4 m long. How tall is the flagpole?

A.3.60 m
B.4.05 m
C.4.50 m
D.5.20 m

Show the answer and why

Answer: B 4.05 m

Why

Set up the proportion: Height / Shadow = 1.8 / 4 = h / 9. Solving for height gives h = (1.8 * 9) / 4 = 16.2 / 4 = 4.05 m.

Question 8

Two similar triangles have areas of 144 cm² and 169 cm². If a side of the larger triangle is 26 cm, what is the length of the corresponding side of the smaller triangle?

A.20 cm
B.22 cm
C.24 cm
D.25 cm

Show the answer and why

Answer: C 24 cm

Why

The ratio of corresponding sides is the square root of the ratio of areas: √(144 / 169) = 12 / 13. The side of the smaller triangle is 26 * (12 / 13) = 24 cm.

Question 9

Abdi is 1.8 meters tall and casts a shadow of 2.4 meters. His younger brother casts a shadow of 2 meters at the same time. How tall is Abdi's brother?

A.1.2 meters
B.1.4 meters
C.1.5 meters
D.1.6 meters

Show the answer and why

Answer: C 1.5 meters

Why

By similarity of triangles formed with the ground: Height / Shadow = 1.8 / 2.4 = h / 2. Solving for h gives h = (1.8 * 2) / 2.4 = 3.6 / 2.4 = 1.5 meters.

Question 10

A photograph measuring 4 cm wide and 5 cm long is enlarged to make a wall mural in a hall in Addis Ababa. If the mural is 120 cm wide, what is the length of the mural?