Special Right Triangles

13.6: Special Right Triangles. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 6 of 63 parts Practice Flashcards

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45-45-90 triangles

Objective: Use equal legs and a √2 hypotenuse multiplier.

Prerequisite skills: right triangles and radicals. An isosceles right triangle has two 45° angles. If each leg is a, the hypotenuse is a√2 by the Pythagorean theorem. If the hypotenuse is known, divide by √2 to find a leg. Common error: doubling a leg to get the hypotenuse. Restrictions and interpretation: Identify which side is opposite 30° or 60°; ratios do not depend on drawing orientation.

Formulas

leg:leg:hypotenuse=1:1:√2

Worked examples

Try itIn a 45-45-90 triangle with leg 24, the hypotenuse is k√2. What is k?Show answer

Both legs equal 24. Pythagoras gives c²=2(576), hence c=24√2, so k=24. Therefore the requested answer is 24.

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45-45-90 triangles

Objective: Use equal legs and a √2 hypotenuse multiplier.

Formulas

leg:leg:hypotenuse=1:1:√2

Worked examples

Try itIn a 45-45-90 triangle with leg 24, the hypotenuse is k√2. What is k?Show answer

Both legs equal 24. Pythagoras gives c²=2(576), hence c=24√2, so k=24. Therefore the requested answer is 24.

30-60-90 triangles

Objective: Relate the short leg, long leg, and hypotenuse.

The side opposite 30° is the short leg s; opposite 60° is s√3; opposite 90° is 2s. Naming the angles fixes correspondence more reliably than a sketch. Common error: treating the side opposite 60° as the short leg. Restrictions and interpretation: Identify which side is opposite 30° or 60°; ratios do not depend on drawing orientation.

Formulas

s:s√3:2s

Worked examples

Try itA 30-60-90 triangle has hypotenuse 48. What is the length of the side opposite 30°?Show answer

The side opposite 30° is half the hypotenuse: 48/2=24. The longer leg would instead be 24√3. Therefore the requested answer is 24.

Recover an area from special sides

Objective: Combine a special-triangle ratio with the area formula.

Once the leg lengths are known, the area is half their product because the legs are perpendicular. A supplied radical length can often be simplified symbolically before calculation. Common error: using the hypotenuse as the perpendicular height. Restrictions and interpretation: Identify which side is opposite 30° or 60°; ratios do not depend on drawing orientation. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$30-60-90 area=(s²√3)/2$

Worked examples

Try itA 30-60-90 triangle has short leg 50. Its area is K√3. What is K?Show answer

The long leg is 50√3. Area=½(50)(50√3)=(2500/2)√3, so K=1250. Therefore the requested answer is 1250.

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