Pythagoras Theorem
Multiple Choice Questions (MCQ)
1. In a right-angled triangle, the square of the hypotenuse is equal to:
A. The sum of all sides
B. The difference of the other two sides
C. The sum of the squares of the other two sides
D. Twice the product of the other two sides
Answer: C. The sum of the squares of the other two sides
2. Which of the following sets of lengths can form a right-angled triangle?
A. 2 cm, 3 cm, 4 cm
B. 5 cm, 12 cm, 13 cm
C. 6 cm, 6 cm, 6 cm
D. 7 cm, 10 cm, 12 cm
Answer: B. 5 cm, 12 cm, 13 cm
Q3. If triangle ABC has ∠B = 90°, and AB = 9 cm, BC = 12 cm, then AC is:
A. 10 cm
B. 15 cm
C. 13 cm
D. 18 cm
Answer: C. 15 cm
Q4. The converse of Pythagoras Theorem is used to:
A. Find the area of a triangle
B. Prove a triangle is equilateral
C. Check if a triangle is right-angled
D. Calculate the perimeter
Answer:C. Check if a triangle is right-angled
Q5. In triangle XYZ, right-angled at Y, XY = 6 cm and YZ = 8 cm. What is the hypotenuse XZ?
A. 9 cm
B. 11 cm
C. 10 cm
D. 12 cm
Answer: C. 10 cm
Short Answer Questions (SAQ)
1. State the Pythagoras Theorem in mathematical form.
Answer: If ∠B = 90° in triangle ABC, then:
AC² = AB² + BC²
2. What is the hypotenuse in a right-angled triangle?
Answer: The side opposite the right angle and the longest side is the hypotenuse.
3. Can the Pythagoras Theorem be used in an obtuse-angled triangle? Why or why not?
Answer: No, it is only valid in right-angled triangles.
Long Answer Questions (LAQ)
1. Prove the Pythagoras Theorem using geometric construction.
Answer:
- Construct triangle ABC with ∠B = 90°.
- Construct squares on all three sides: AB², BC², and AC².
- Calculate areas of the smaller squares (AB² and BC²).
- Show that the sum of their areas equals the area of the square on AC (hypotenuse).
- Hence, prove: AC² = AB² + BC²
2. A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Verify whether it is a right-angled triangle.
Answer: Check:
7² + 24² = 49 + 576 = 625
25² = 625
Since the square of the longest side equals the sum of the squares of the other two: Yes, it is a right-angled triangle (Right angle opposite 25 cm side).