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

Students were asked over to answer a question at schools and to indicate what is most important for them to succeed. One that response stood out from the rest was practice. Successful persons commonly are not born successful; they become successful through hard work and perseverance. If you would like to accomplish your goals, keep this in mind! as follows some question and answer examples that you will be able to utilize to develop your knowledge and gain insight that will assist you to continue your school studies.

Question:

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

Answer:

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

Learn more about the AA similarity postulate and geometric mean here:

They could possibly hopefully help the student deal with the question by working with the questions and answer examples. You could simply then have a discussion with your classmate and continue the school learning by studying the subject altogether.

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