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Similar Triangles

Calculate missing sides using scale factors

Triangle 1 (Original)

Triangle 2 (Similar)

How To Solve Similar Triangles

Step 1: Verify Triangle Similarity

Confirm triangles are similar (AA, SAS, or SSS similarity criterion).

Why: The proportionality only holds for similar triangles, not arbitrary pairs.

Step 2: Identify Corresponding Sides

Determine which sides correspond between the two triangles (a↔a', b↔b', c↔c').

Why: Incorrect correspondence invalidates all subsequent calculations.

Step 3: Set Up Ratios from Known Data

Use at least one pair of known corresponding sides to establish the scale factor k.

Why: The scale factor is constant for all corresponding sides of similar figures.

Step 4: Calculate Unknown Sides

Multiply each known side of Triangle 1 by k to find the corresponding side in Triangle 2.

Why: All corresponding sides must scale by the same factor for similarity to hold.

Step 5: Verify Proportionality

Check that all calculated ratios equal k and that all values are geometrically valid.

Why: Discrepancies indicate calculation errors or non-similar triangles.

Detailed Example

Scenario: Triangle 1 has sides 3, 4, 5. Triangle 2 is similar with one side = 9. Find other sides.
Step 1 - Verify Similarity: Both are right triangles (3-4-5 is a Pythagorean triple).
Step 2 - Identify Correspondence: Side 3 of Triangle 1 corresponds to side 9 of Triangle 2.
Step 3 - Find Scale Factor: k = 9/3 = 3 (Triangle 2 is 3 times larger).
Step 4 - Calculate Sides: a' = 3 × 3 = 9, b' = 4 × 3 = 12, c' = 5 × 3 = 15.
Step 5 - Verify: 9/3 = 3, 12/4 = 3, 15/5 = 3. All ratios = k ✓ Check: 9² + 12² = 81 + 144 = 225 = 15² ✓
Verification: Triangle 2 is a 9-12-15 right triangle (3 times the 3-4-5).
Result: Triangle 2 has sides 9, 12, and 15 cm.
Interpretation: Similar triangles scale proportionally; the scale factor applies uniformly to all dimensions.

Geometric Principle

Similar triangles have proportional corresponding sides. The ratio between any two corresponding sides is called the scale factor (k).a'/a = b'/b = c'/c = k

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