Understanding Similar Triangles

Year 8 Mathematics — AC9M8SP01

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📖 Part 1: Key Concepts

Similar triangles are triangles that have the same shape but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are in proportion.

Three conditions (criteria) for triangle similarity:

• AA (Angle-Angle): Two pairs of corresponding angles are equal.

• SSS (Side-Side-Side): All three pairs of corresponding sides are in the same ratio.

• SAS (Side-Angle-Side): Two pairs of corresponding sides are in proportion AND the included angle is equal.

• RHS (Right Angle-Hypotenuse-Side): The Right Angles are equal AND the hypotenuse (side) and another corresponding side are in proportion.

1. Complete the sentence using the words in the box: [equal / proportion / shape / size]

In similar triangles, corresponding angles are __________ and corresponding sides are in __________. The triangles have the same __________ but may differ in __________.

2. Match each similarity condition to its correct description by writing the letter in the blank.

A. AA

B. SSS

C. SAS

___ All three side ratios are equal.

___ Two angles are equal.

___ Two sides are proportional and the included angle is equal.

3. Which properties stay the same and which change when a triangle is enlarged? Tick all that apply.

Stays the same:

Angle sizes

Side lengths

Overall shape

Area

Changes:

Angle sizes

Side lengths

Overall shape

Area

✏️ Part 2: Problem Solving

4. Triangle ABC is similar to Triangle DEF. The sides of Triangle ABC are 3 cm, 4 cm, and 5 cm. Triangle DEF has been enlarged by a scale factor of 3. Find the three side lengths of Triangle DEF.

Side lengths of Triangle DEF: __________ cm, __________ cm, __________ cm

5. Two similar triangles have the following measurements. Find the missing side length x.

Triangle 1 has sides: 6 cm, 9 cm, 12 cm
Triangle 2 has sides: 4 cm, 6 cm, x cm

x = __________ cm

6. Determine the similarity condition (AA, SSS, or SAS) for each pair of triangles below.

(a) Triangle PQR has angles 45°, 65°, 70°. Triangle XYZ has angles 45°, 65°, 70°.

Similarity condition: __________

(b) Triangle ABC has sides 4 cm, 6 cm, 8 cm. Triangle DEF has sides 2 cm, 3 cm, 4 cm.

Similarity condition: __________

7. 🌟 Real-Life Application: Estimating Height Using Shadows

On a sunny afternoon, Mia notices that her 1.6 m tall friend casts a shadow of 2 m on the ground. At the same time, a nearby tree casts a shadow of 7 m. Using similar triangles, estimate the height of the tree. Show your working.

Height of the tree ≈ __________ m

8. On the grid below, draw a triangle with vertices at (1, 1), (3, 1), and (1, 4). Then draw an enlargement of the triangle using a scale factor of 2 from the origin. Label both triangles.

What do you notice about the angles of the two triangles? ___________________________________