Similar Triangles UPCAT Reviewer: Lesson and Practice

TEACHER ABI UPCAT MATHEMATICS

Similar Triangles

Match corresponding parts first, then apply one consistent scale factor.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Similar Triangles

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Match corresponding parts before writing any proportion

Similar triangles have the same shape: corresponding angles are equal and corresponding sides are proportional. The order of the names records the correspondence. If △ABC∼△DEF, then A↔D, B↔E, C↔F, and ABDE=BCEF=ACDF.

lengths and perimeters scale by kareas scale by k²volumes of similar solids scale by k³

Length scale

larger side ÷ corresponding smaller side

Area scale

If lengths scale by k, areas scale by k².

DO IT FAST

Match first; calculate second

Problem: Two triangles are similar. A 6-cm side in the smaller triangle corresponds to 18 cm in the larger. What corresponds to a 5-cm side?

Recognize: The larger triangle is an enlargement of the smaller one. Since 6 cm became 18 cm, each corresponding side was multiplied by 3.

Corresponding sides occupy matching positions and use the same scale factor.

Do it fast: Scale factor = 18÷6=3. Apply that same factor to the matching 5-cm side: 5×3=15 cm.

Answer: 15 cm.

Before calculating, confirm that the sides correspond—usually by matching the equal angles opposite them.

Why it works

Similar triangles are one consistent enlargement or reduction. Once the correct scale factor is known, it applies to every corresponding length.

WORKED EXAMPLES

Five forms you should recognize

1. Find a corresponding length

If 6 cm corresponds to 18 cm, the scale factor is 186=3. A matching 5-cm side becomes:

5(3)=15 cm

2. Use the naming order

For △ABC∼△DEF, side BC corresponds to EF because B↔E and C↔F. Write ratios in the same direction.

3. Parallel segment

If DE∥BC inside △ABC, then △ADE∼△ABC. With AD=4, AB=10, and AE=5:

410=5AC → AC=12.5

4. Area relationship

A length scale factor of 3 gives an area factor of 3²=9. A smaller area of 8 cm² becomes:

8(9)=72 cm²

5. Indirect height

A 1.5-m pole casts a 2-m shadow while a tree casts a 12-m shadow:

1.52=h12 → h=9 m

COMMON TRAPS

Check before you commit

• Matching sides by visual position
• Reversing one ratio
• Adding the scale factor
• Using k instead of k² for area
• Assuming similar means congruent
• Trusting a diagram’s scale

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Similar Triangles FAQ

Why are there two mastery sets?

A different five-question set appears after an unsuccessful attempt so memorizing one set does not verify mastery.

Does 5/5 mean I never need this topic again?

It verifies this competency today; revisit it later through mixed practice.

RELATED COMPETENCIES

Continue your mathematics review.

Ratio and ProportionCongruenceAreaPythagorean Theorem

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