Math Problem Statement

Determine the lengths of the unknown sides in the following pairs of similar triangles.

Solution

To determine the lengths of the unknown sides in the given pairs of similar triangles, we can use the property that corresponding sides of similar triangles are proportional.

Analyzing the Triangles:

  1. Larger Triangle:

    • Sides: xx, 6, 9
    • Given side ratios: x2=6y=93\frac{x}{2} = \frac{6}{y} = \frac{9}{3}
  2. Smaller Triangle:

    • Sides: 2, yy, 3

Step 1: Use the proportion based on the sides that are already known.

93=3\frac{9}{3} = 3

This ratio applies to all corresponding sides.

Step 2: Calculate xx using the ratio.

x2=3    x=2×3=6\frac{x}{2} = 3 \implies x = 2 \times 3 = 6

Step 3: Calculate yy using the ratio.

6y=3    y=63=2\frac{6}{y} = 3 \implies y = \frac{6}{3} = 2

Final Values:

  • x=6x = 6
  • y=2y = 2

The unknown sides are x=6x = 6 and y=2y = 2.

Would you like further details or have any questions about this solution?

Related Questions:

  1. How do you determine if two triangles are similar?
  2. Can similar triangles have sides that are not proportional?
  3. What is the significance of the scale factor in similar triangles?
  4. How does the Pythagorean Theorem relate to similar triangles?
  5. What happens if one of the triangles in the pair is a right triangle?

Tip:

When working with similar triangles, always ensure that you correctly identify corresponding sides before setting up proportions.

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Math Problem Analysis

Mathematical Concepts

Similar Triangles
Proportionality
Triangle Properties

Formulas

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Theorems

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Suitable Grade Level

High School