Question 1: Calculate the shorter side 'a' in a right-angled triangle where the hypotenuse 'c' is 10 units, and the longer leg 'b' is 6 units.

Answer: 8

Question 2: A triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Find the other shorter side 'a'.

Answer: 12

Question 3: Using the formula a² = c² - b², determine if the calculation 7² = 10² - 4² is correct. Explain your reasoning.

Answer: No, because 7²=49 while 10²-4²=100-16=84, which are not equal.

Question 4: A student mistakenly uses the formula a² = c² + b² instead of a² = c² - b². If the hypotenuse is 15 units and the other side is 9 units, what is the student's calculated value for 'a'? Is it correct?

Answer: 24 (incorrect)

Question 5: Construct a triangle on the grid such that the hypotenuse is 12 units and one leg is 5 units. Use the formula a² = c² - b² to find the other leg 'a'.

Answer: 11

Question 6: A right triangle has sides measuring 9 units and 12 units. Verify whether these can be the legs or the hypotenuse using a² = c² - b².

Answer: Yes, if 15 is the hypotenuse; 9²+12²=81+144=225; √225=15.

Question 7: A rectangle has a diagonal of 20 units. If one side is 12 units, find the length of the adjacent side using a² = c² - b².

Answer: 16

Question 8: Explain why using a² = c² + b² instead of a² = c² - b² in a right triangle leads to incorrect results.

Answer: Because in a right triangle, the Pythagorean theorem states that the square of the hypotenuse equals the sum of the squares of the legs, not the difference.

Question 9: A ladder leans against a wall, with the hypotenuse (ladder length) of 25 meters. The foot of the ladder is 7 meters from the wall. Find the height at which the ladder touches the wall.

Answer: 24

Question 10: Identify the mistake in this solution: The student claims that in a right triangle with hypotenuse 10 units and one leg 6 units, the other leg is calculated as √(10² + 6²) = √136 ≈ 11.66.

Answer: The mistake is adding squares instead of subtracting; the correct formula is a² = c² - b².

Question 11: Challenge: For a right triangle, if the two shorter sides are 8 units and 15 units, verify whether the longer side can be the hypotenuse using a² = c² - b².

Answer: Yes, c= √(8²+15²)=√(64+225)=√289=17, so hypotenuse is 17 units.