Pythagorean Theorem MCQ Test — Class 8 CBSE

Solution:

The Pythagorean theorem establishes a fundamental relationship between the three sides of a right-angled triangle.

It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs). — a² + b² = c²

Therefore, its application is strictly limited to right-angled triangles.

Solution:

The Pythagorean theorem states a² + b² = c², where 'c' is the hypotenuse (the longest side).

Given two sides of a right-angled triangle, the unknown side could be either the hypotenuse or one of the legs.

If 5 cm and 12 cm are the legs, then the equation should be 5² + 12² = x² (where x is the hypotenuse). In this case, x would be √(25+144) = √169 = 13 cm.

If 12 cm is the hypotenuse and 5 cm is one leg, then the equation 5² + x² = 12² is correct for finding the other leg. Ravi's mistake is in not considering the possibility that 12 cm could be the hypotenuse if x is the other leg, implying x is smaller than 12.

Solution:

In a right-angled triangle, the angle measuring 90° is called the right angle.

The side directly opposite to the right angle is defined as the hypotenuse.

Since the right angle is at Q, the side opposite to angle Q is PR. Therefore, PR is the hypotenuse.

Solution:

A Pythagorean triplet consists of three positive integers that satisfy the Pythagorean theorem.

If 'a', 'b', and 'c' are natural numbers, they form a Pythagorean triplet if a² + b² = c².

This means they can be the sides of a right-angled triangle, where 'c' is the hypotenuse.

Solution:

When a diagonal is drawn in a rectangle, it forms two right-angled triangles.

The length and width of the rectangle act as the two legs (perpendicular and base) of the right-angled triangle, and the diagonal is the hypotenuse.

Using the Pythagorean theorem: Diagonal² = Length² + Width².

Diagonal² = 80² + 60² = 6400 + 3600 = 10000. So, Diagonal = √10000 = 100 meters.

Solution:

Let the hypotenuse (c) be 13 cm and one leg (a) be 5 cm. Let the other leg be 'b'.

According to the Pythagorean theorem: a² + b² = c².

Substitute the known values: 5² + b² = 13².

Calculate: 25 + b² = 169. So, b² = 169 - 25 = 144. Therefore, b = √144 = 12 cm.

Solution:

The Pythagorean theorem states that if a triangle is right-angled, then a² + b² = c² (where c is the hypotenuse).

If a² + b² < c², the triangle is obtuse-angled (the angle opposite side c is obtuse).

If a² + b² > c², the triangle is acute-angled (all angles are acute, specifically the angle opposite the longest side c is acute).

Given p² + q² > r², with r being the longest side, it implies that the angle opposite to side r is acute. Thus, the triangle is an acute-angled triangle.

Solution:

We need to check which set satisfies a² + b² = c² (where c is the largest number).

For (2, 3, 4): 2² + 3² = 4 + 9 = 13. But 4² = 16. Since 13 ≠ 16, it's not a triplet.

For (4, 5, 6): 4² + 5² = 16 + 25 = 41. But 6² = 36. Since 41 ≠ 36, it's not a triplet.

For (6, 7, 8): 6² + 7² = 36 + 49 = 85. But 8² = 64. Since 85 ≠ 64, it's not a triplet.

For (7, 24, 25): 7² + 24² = 49 + 576 = 625. And 25² = 625. Since 625 = 625, it IS a Pythagorean triplet.

Solution:

The ladder, the wall, and the ground form a right-angled triangle.

The length of the ladder (17 m) is the hypotenuse (c).

The distance of the foot of the ladder from the wall (8 m) is one leg (a).

The height the ladder reaches on the wall (h) is the other leg (b).

Using the Pythagorean theorem: a² + b² = c² => 8² + h² = 17².

64 + h² = 289. So, h² = 289 - 64 = 225. Therefore, h = √225 = 15 m.

Solution:

We need to find a pair of leg lengths (a, b) such that a² + b² = c² where c = 10 cm. So, a² + b² = 10² = 100.

Option A: 3² + 4² = 9 + 16 = 25. This is not 100.

Option B: 6² + 8² = 36 + 64 = 100. This matches the hypotenuse squared.

Option C: 5² + 5² = 25 + 25 = 50. This is not 100.

Option D: 7² + 7² = 49 + 49 = 98. This is not 100.

Thus, the legs could be 6 cm and 8 cm.