CAT Geometry: 30 Triangles & Pythagorean Triplets Practice Questions with Solutions
Q1–10 of 30What is the hypotenuse of a right-angled triangle whose legs are 24 cm and 70 cm?
Explanation: Factoring out 2 gives 2 × (12, 35). The primitive base triplet is (12, 35, 37), so the hypotenuse is 2 × 37 = 74 cm.
An odd integer n = 13 is the shortest side of a primitive Pythagorean triplet (a, b, c). Find the hypotenuse c.
Explanation: For an odd integer n, hypotenuse c = (n² + 1) / 2 = (13² + 1) / 2 = 170 / 2 = 85. (The other leg is (13² − 1) / 2 = 84.)
An even integer n = 16 is the shortest side of a primitive Pythagorean triplet. What is the length of the hypotenuse?
Explanation: For an even number n, let k = n / 2 = 8. The hypotenuse is k² + 1 = 8² + 1 = 65, giving the primitive triplet (16, 63, 65).
A right-angled triangle has legs of lengths 15 cm and 36 cm. What is its inradius?
Explanation: (15, 36, c) = 3 × (5, 12, 13) ⟹ c = 39 cm. Inradius r = (a + b − c) / 2 = (15 + 36 − 39) / 2 = 6 cm.
In a right triangle with legs 25 cm and 60 cm, find the length of the altitude drawn to the hypotenuse.
Explanation: (25, 60) = 5 × (5, 12) ⟹ Hypotenuse c = 65 cm. Altitude h = (a × b) / c = (25 × 60) / 65 = 300/13 cm.
The sides of a right-angled triangle are in the ratio 8:15:17. If its perimeter is 160 cm, what is its area?
Explanation: Perimeter = (8 + 15 + 17)x = 40x = 160 ⟹ x = 4. Legs are 32 cm and 60 cm. Area = ½ × 32 × 60 = 960 cm².
In a primitive Pythagorean triplet (a, b, c), the hypotenuse c and longer leg b differ by 1. If the shortest leg is 9, what is the value of b?
Explanation: c² − b² = 9² ⟹ (c − b)(c + b) = 81. With c − b = 1, c + b = 81 ⟹ 2b = 80 ⟹ b = 40.
A right-angled triangle has a hypotenuse of 29 cm and a perimeter of 70 cm. Find the length of its shortest leg.
Explanation: a + b = 70 − 29 = 41. Since 20 + 21 = 41 and 20² + 21² = 29², the triplet is (20, 21, 29), and the shortest leg is 20 cm.
How many non-congruent right-angled triangles with integer sides have a hypotenuse of exactly 50 cm?
Explanation: Triangles are formed by scaling base triplets whose hypotenuses divide 50: (3, 4, 5) × 10 = (30, 40, 50) and (7, 24, 25) × 2 = (14, 48, 50). No other primitive hypotenuse divides 50.
An integer-sided right triangle has an area of 30 cm² and a perimeter of 30 cm. What is the length of its hypotenuse?
Explanation: Semi-perimeter s = 15 ⟹ Inradius r = Area / s = 30 / 15 = 2. Using r = (a + b − c) / 2 ⟹ a + b − c = 4. Since a + b + c = 30, subtracting yields 2c = 26 ⟹ c = 13 cm.