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Q1. What is the area of an isosceles triangle with a base of 10 cm and equal sides of 13 cm?
Solution:
Using Heron's formula, s = (10 + 13 + 13) / 2 = 18. Area = √(s(s-a)(s-b)(s-c)) = √(18(18-10)(18-13)(18-13)) = √(18*8*5*5) = 60 cm².
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Q2. A triangle has sides of lengths 9 cm, 12 cm, and 15 cm. Is it a right triangle?
Solution:
Yes, because 9² + 12² = 81 + 144 = 225 = 15².
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Q3. If a triangle has an area of 36 cm² and a base of 9 cm, what is its height?
Solution:
Area = 1/2 * base * height. 36 = 1/2 * 9 * height. Height = (36 * 2) / 9 = 8 cm.
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Q4. What is the length of the altitude to the base of a triangle with an area of 50 cm² and a base of 10 cm?
Solution:
Area = 1/2 * base * height. 50 = 1/2 * 10 * height. Height = (50 * 2) / 10 = 10 cm.
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Q5. What is the height of a triangle with an area of 60 cm² and a base of 12 cm?
Solution:
Area = 1/2 * base * height. Therefore, 60 = 1/2 * 12 * height. Height = 60 / 6 = 10 cm.
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Q6. In a right triangle, if one leg is 6 cm and the other leg is 8 cm, what is the area?
Solution:
Area = 1/2 * leg1 * leg2 = 1/2 * 6 * 8 = 24 cm².
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Q7. If the sides of a triangle are 7 cm, 24 cm, and 25 cm, what is its area?
Solution:
Using Heron's formula, s = (7 + 24 + 25)/2 = 28. Area = √(s(s-a)(s-b)(s-c)) = √(28(28-7)(28-24)(28-25)) = √(28*21*4*3) = 84 cm².
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Q8. If the lengths of two sides of a triangle are 5 cm and 12 cm, and the included angle is 90 degrees, what is the area?
Solution:
Area = 1/2 * side1 * side2 = 1/2 * 5 * 12 = 30 cm².
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Q9. A triangle has an area of 48 cm² and a base of 8 cm. What is the height?
Solution:
Area = 1/2 * base * height. Therefore, 48 = 1/2 * 8 * height. Solving gives height = 48 / 4 = 12 cm.
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Q10. A triangle has vertices at (0, 0), (4, 0), and (0, 3). What is its area?
Solution:
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6 cm².
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