Right Triangles and Trigonometry — Free Geometry Review Games.
This unit covers Pythagorean theorem, special right triangles and sine cosine tangent — essential concepts for Geometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers Pythagorean theorem, special right triangles and sine cosine tangent — essential concepts for Geometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Pythagorean Theorem
The Pythagorean Theorem states that in any right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle). Students must be able to find a missing side given the other two, and also use the converse to determine whether a triangle is a right triangle.
Key Points
- c is always the hypotenuse — the longest side, opposite the 90° angle
- To find a leg: a² = c² − b² (subtract, then square root)
- Converse: if a² + b² = c², the triangle is a right triangle
- Common Pythagorean triples to memorize: 3-4-5, 5-12-13, 8-15-17 (and their multiples)
A right triangle has legs of length 6 and 8. Find the hypotenuse.
Substitute into a² + b² = c²: 6² + 8² = c², giving 36 + 64 = 100. Taking the square root, c = 10. This is also a 3-4-5 triple scaled by 2, so recognizing the pattern saves time on exams.
2 Special Right Triangles
There are two special right triangles with fixed side ratios that students must memorize: the 45-45-90 triangle (ratio 1 : 1 : √2) and the 30-60-90 triangle (ratio 1 : √3 : 2). These appear frequently on exams because they allow exact answers without a calculator.
Key Points
- 45-45-90: legs are equal; hypotenuse = leg × √2
- 30-60-90: short leg opposite 30°; long leg = short leg × √3; hypotenuse = short leg × 2
- To work backwards: if the hypotenuse is given, divide by √2 (45-45-90) or by 2 (30-60-90) to find the short leg
- Rationalize denominators when simplifying (e.g., 5/√2 = 5√2/2)
A 30-60-90 triangle has a hypotenuse of 14. Find the length of both legs.
The short leg (opposite 30°) equals half the hypotenuse: 14 ÷ 2 = 7. The long leg (opposite 60°) equals the short leg times √3: 7√3. The three sides are 7, 7√3, and 14.
3 Sine, Cosine, And Tangent
Sine, cosine, and tangent are ratios that relate an acute angle in a right triangle to two of its sides. Students must know the mnemonic SOH-CAH-TOA, be able to set up and solve for missing sides, and use inverse trig functions to find missing angles.
Key Points
- sin(θ) = opposite / hypotenuse; cos(θ) = adjacent / hypotenuse; tan(θ) = opposite / adjacent
- To find a missing side: set up the correct ratio, then solve algebraically (multiply or divide)
- To find a missing angle: use the inverse function — θ = sin⁻¹(ratio), cos⁻¹(ratio), or tan⁻¹(ratio)
- Label sides relative to the given angle — 'opposite' and 'adjacent' change depending on which angle you use
In a right triangle, the angle θ = 35° and the hypotenuse = 20. Find the side opposite θ.
Use sine because the problem involves opposite and hypotenuse: sin(35°) = opposite / 20. Multiply both sides by 20: opposite = 20 × sin(35°) ≈ 20 × 0.574 ≈ 11.47. On an exam, set up the equation first before reaching for a calculator.
Questions, answered.
What is Right Triangles and Trigonometry?
Right Triangles and Trigonometry is Unit 8 of Geometry, covering Pythagorean theorem, special right triangles and sine cosine tangent.
How to study for Geometry Unit 8?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.