# Unit 8 right triangles and trigonometry homework 3 similar right triangles and geometric mean

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## Question:

Unit 8 right triangles and trigonometry homework 3 similar right triangles and geometric mean The right triangles that have an altitude which forms two right triangles

are similar to the two right triangles formed.

Responses:

1. ΔLJK ~ ΔKJM

ΔLJK ~ ΔLKM

ΔKJM ~ ΔLKM

2. ΔYWZ ~ ΔZWX

ΔYWZ ~ ΔYZW

ΔZWX ~ ΔYZW

3. x = 4.8

4. x ≈ 14.48

5. x ≈ 11.37

6. G.M. = 12·√3

7. G.M. = 6·√5

### What condition guarantees the similarity of the right triangles?

1. ∠LMK = 90° given

∠JMK + ∠LMK  = 180° linear pair angles

∠JMK = 180° – 90° = 90°

∠JKL ≅ ∠JMK All 90° angles are congruent

∠LJK ≅ ∠LJK reflexive property

• ΔLJK is similar to ΔKJM by Angle–Angle, AA, similarity postulate

∠JLK ≅ ∠JLK by reflexive property

• ΔLJK is similar to ΔLKM by AA similarity

By the property of equality for triangles that have equal interior angles, we have;

• ΔKJM ~ ΔLKM

2. ∠YWZ ≅ ∠YWZ by reflexive property

∠WXZ ≅ ∠YZW all 90° angle are congruent

• ΔYWZ is similar to ΔZWX, by AA similarity postulate
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∠XYZ ≅ ∠WYZ by reflexive property

∠YXZ ≅ ∠YZW all 90° are congruent

• ΔYWZ is similar to ΔYZW by AA similarity postulate

Therefore;

• ΔZWX ~ ΔYZW

3. The ratio of corresponding sides in similar triangles are equal

From the similar triangles, we have; 8 × 6 = 10 × x

48 = 10·x

• 3. From the similar triangles, we have; 20 × 21 = x × 29

420 = 29·x

• 4. From the similar triangles, we have; 20 × 48 = 52 × x

• 5. From the similar triangles, we have; 13.2 × 22.4 = 26 × x

• 6. The geometric mean, G.M. is given by the formula; The geometric mean of 16 and 27 is therefore;

• • The geometric mean of 16 and 27 is 12·√3

7. The geometric mean of 5 and 36 is found as follows; • The geometric mean of 5 and 36 is 6·√5 