Math Problem Statement

Prove 9th one sec main

Prove 9th one sec main

Solution

Let's solve the second part of Question 9, which is to prove the following:

CAPA=BCMP\frac{CA}{PA} = \frac{BC}{MP}

Problem Setup:

We are given that ABC\triangle ABC and AMP\triangle AMP are right-angled triangles. ABC\triangle ABC is right-angled at BB, and AMP\triangle AMP is right-angled at MM. From part (i), we already know that these two triangles are similar by the AA similarity criterion.

Solution Steps:

  1. Using the similarity from part (i): Since ABCAMP\triangle ABC \sim \triangle AMP (proven in part i), the corresponding sides of the triangles will be in proportion.

  2. Corresponding sides: From the similarity, the corresponding sides of the two triangles are:

    • CACA (the hypotenuse of ABC\triangle ABC) corresponds to PAPA (the hypotenuse of AMP\triangle AMP).
    • BCBC (the side opposite A\angle A in ABC\triangle ABC) corresponds to MPMP (the side opposite A\angle A in AMP\triangle AMP).
  3. Proportionality of corresponding sides: Since the triangles are similar, the ratio of the corresponding sides will be equal: CAPA=BCMP\frac{CA}{PA} = \frac{BC}{MP}

Thus, the given equation is proven based on the similarity of the two triangles.


Would you like to proceed with another part of this problem or discuss further details?

Here are some related questions for you:

  1. How can we use Pythagoras' theorem to calculate the unknown sides in right-angled triangles?
  2. What happens to the ratios of corresponding sides when the triangles are congruent instead of similar?
  3. How can similarity in triangles be used to solve real-world measurement problems?
  4. What other properties of the triangles can you infer once the similarity is established?
  5. How does the AA similarity criterion differ from the SSS and SAS criteria?

Tip: In right-angled triangles, always look for the possibility of similarity through AA when one common angle and one right angle are given.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Similarity
Proportionality

Formulas

CA / PA = BC / MP (proportionality of corresponding sides in similar triangles)

Theorems

AA Similarity Criterion (Angle-Angle similarity)
Corresponding sides of similar triangles are proportional

Suitable Grade Level

Grade 10-11