Math Problem Statement

D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that CA^2 = CB · CD.

Solution

I see the mathematical problems in the image you uploaded. Let’s focus on the 13th problem first:

Problem 13:
"D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that CA2=CBCDCA^2 = CB \cdot CD."

This is a classic problem that involves properties of triangles, particularly the angle bisector theorem and the use of similar triangles. Here is a step-by-step approach to prove this statement:

Proof:

  1. Identify similar triangles:
    Since ADC=BAC\angle ADC = \angle BAC, triangles ΔADC\Delta ADC and ΔABC\Delta ABC are similar by the AA (Angle-Angle) similarity criterion, because they also share the angle A\angle A.

  2. Apply the similarity ratio:
    Since ΔADCΔABC\Delta ADC \sim \Delta ABC, corresponding sides are proportional. Hence, we can write: CACB=CDCA\frac{CA}{CB} = \frac{CD}{CA}

  3. Cross-multiply the proportion: By cross-multiplying the terms, we get: CA2=CBCDCA^2 = CB \cdot CD

Thus, the proof is complete. This relation shows how the square of one side (in this case, CACA) relates to the product of two other segments on the opposite side.


Would you like further clarification or additional details on any step?
Here are five related questions to expand your understanding:

  1. What is the AA (Angle-Angle) criterion for similar triangles?
  2. How does the Angle Bisector Theorem relate to triangle similarity?
  3. Can this result be generalized to other types of triangles?
  4. What are other methods to prove triangle similarity?
  5. How can this result be applied to find specific lengths in a triangle?

Tip: Understanding triangle similarity helps with solving problems in both geometry and trigonometry, especially when working with proportions and relationships between sides.

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Math Problem Analysis

Mathematical Concepts

Triangle similarity
Proportionality of sides
Geometric proof

Formulas

CA^2 = CB · CD

Theorems

Angle-Angle (AA) similarity
Angle Bisector Theorem

Suitable Grade Level

Grades 9-11