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1. Learning Objectives
- Identify the hypotenuse, opposite, and adjacent sides of a right-angled triangle.
- Apply the Pythagorean Theorem to find missing side lengths.
- Recognize common Pythagorean triples.
2. Key Notes
A right-angled triangle has one angle equal to 90°. The side opposite the 90° angle is the hypotenuse (the longest side).
Pythagorean Theorem: a² + b² = c²
Where a and b are the legs, and c is the hypotenuse.
Side Identification: Relative to a chosen angle θ:
- Opposite: The side across from θ.
- Adjacent: The side next to θ (not the hypotenuse).
Click the button below each question to reveal the explanation.
1. If the legs of a triangle are 3 and 4, find the hypotenuse.
3² + 4² = 9 + 16 = 25. √25 = 5. Ans: 5
2. Find the missing leg if the hypotenuse is 13 and one leg is 5.
13² - 5² = 169 - 25 = 144. √144 = 12. Ans: 12
3. Which side is always the longest in a right-angled triangle?
The side opposite the 90° angle. Ans: Hypotenuse
4. Is a triangle with sides 6, 8, and 10 a right-angled triangle?
Check: 6² + 8² = 36 + 64 = 100. 10² = 100. Ans: Yes
5. In ΔABC, if ∠C = 90°, which side is the hypotenuse?
The side opposite angle C is side c (or side AB). Ans: AB
6. Find the hypotenuse of an isosceles right triangle with leg length 1.
1² + 1² = 2. Hypotenuse = √2. Ans: √2
7. What is the value of 8² + 15²?
64 + 225 = 289. Note: √289 = 17. Ans: 289
8. If hypotenuse = 10 and leg = 6, find the other leg.
10² - 6² = 100 - 36 = 64. √64 = 8. Ans: 8
9. True/False: The Pythagorean Theorem works for all triangles.
False. It only works for right-angled triangles. Ans: False
10. Find the distance between (0,0) and (3,4) using the theorem.
Distance = √(3² + 4²) = √25 = 5. Ans: 5
11. A ladder 5m long leans against a wall. Its foot is 3m from the wall. How high does it reach?
Height² = 5² - 3² = 16. Height = 4m. Ans: 4m
12. If a triangle has sides 7, 24, and 25, is it right-angled?
7² + 24² = 49 + 576 = 625. 25² = 625. Ans: Yes
13. What is the area of a right triangle with legs 6 and 8?
Area = 0.5 × base × height = 0.5 × 6 × 8 = 24. Ans: 24
14. If the opposite side to angle θ is 7 and the hypotenuse is 25, what is the adjacent side?
Adj² = 25² - 7² = 625 - 49 = 576. √576 = 24. Ans: 24
15. What do we call a set of three integers that satisfy a² + b² = c²?
Commonly known as 3-4-5, 5-12-13, etc. Ans: Pythagorean Triple
16. Find the diagonal of a rectangle with sides 5 and 12.
Diagonal = √(5² + 12²) = √169 = 13. Ans: 13
17. In a 45°-45°-90° triangle, if the legs are 5, what is the hypotenuse?
Hypotenuse = leg × √2 = 5√2. Ans: 5√2
18. Solve for x: x² + 20² = 29²
x² = 841 - 400 = 441. √441 = 21. Ans: 21
19. If the hypotenuse is 2 and one leg is √3, find the other leg.
Other leg² = 2² - (√3)² = 4 - 3 = 1. Ans: 1
20. What is the sum of the non-right angles in a right triangle?
Triangle sum is 180°. 180° - 90° = 90°. Ans: 90°