Exercises
Put your knowledge of the Pythagorean Theorem to the test with practical right-triangle questions. Solve for hypotenuses and missing legs, identify number sets that form right triangles, work with 45-45-90 triangles, and calculate triangle area. You will also check whether given side lengths satisfy a² + b² = c². This quiz covers classic examples such as 3-4-5 triangles and challenges you to apply the theorem accurately in different situations.
Answer the questions below and check the explanation for each answer.
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To find the length of the hypotenuse in a right-angled triangle, use the Pythagorean theorem: a² + b² = c², where c is the hypotenuse. For sides measuring 3 cm and 4 cm, the calculation is: 3² + 4² = c².
So, 9 + 16 = c², resulting in c² = 25, which gives c = 5 cm.
In a right-angled triangle, use the Pythagorean theorem: a² + b² = c², where c is the hypotenuse. Given one side (b) is 8 cm and the hypotenuse (c) is 10 cm, calculate the other side (a):
a² + 8² = 10²
a² + 64 = 100
a² = 36
a = 6
The length of the other side is 6 cm.
A set of numbers represents the sides of a right-angled triangle if they satisfy the Pythagorean theorem: a² + b² = c², where 'c' is the hypotenuse. Option 1: 6² + 8² = 36 + 64 = 100, and 10² = 100, so it's a right triangle. Option 2: 5² + 5² = 25 + 25 = 50, not equal to 5² = 25, so not a right triangle. Option 3: 5² + 12² = 25 + 144 = 169, and 13² = 169, so it's a right triangle.
In a right triangle with one angle of 45 degrees, it is an isosceles right triangle, meaning the two legs are equal. The ratio of the sides in a 45-45-90 triangle is 1:1:√2. If the hypotenuse is 10√2, each leg is 10 because the legs are equal to the hypotenuse divided by √2 (hypotenuse/√2 = leg).
The area A of a right-angled triangle can be calculated using the formula: A = (base * height) / 2. Here, the legs of the triangle are the base and height, which are 6 cm and 8 cm, respectively. Thus, the area is (6 * 8) / 2 = 24 cm².
To check if the lengths can form a right-angled triangle, apply the Pythagorean theorem: a^2 + b^2 = c^2, where c is the hypotenuse. Here, test if 6^2 = 4^2 + 5^2:
36 ≠ 16 + 25 (36 ≠ 41).
The lengths 4 cm, 5 cm, and 6 cm fail to satisfy the Pythagorean theorem, so they cannot form a right-angled triangle.
To find the area, identify the other leg using the Pythagorean theorem: a² + 5² = 13². Solve for a: a² + 25 = 169 → a² = 144 → a = 12. The area is (1/2) × base × height → (1/2) × 5 × 12 = 30 cm². Correct your choice to the one that matches: 30 cm² is not listed, hence the area is assessed incorrectly. Options errors lead us to choose closest or defined answer with recalculation.
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