Practice Questions
Concept Explanation
A triangle has three sides and three interior angles summing to 180°. Types include equilateral, isosceles, scalene, and right-angled triangles.
Key Formulas
- Area = ½ × base × height
- Heron’s formula: Area = √(s(s − a)(s − b)(s − c)), s = semi-perimeter
- Pythagoras: a² + b² = c² (c = hypotenuse)
- Equilateral triangle: height = (√3/2) × side; area = (√3/4) × side²
Special Triangles
In a 30°-60°-90° triangle, sides are in ratio 1 : √3 : 2 (opposite 30°, 60°, 90°).
Example
Right triangle with legs 3 cm and 4 cm: hypotenuse = 5 cm, area = 6 sq cm.
Common Trap
In Heron’s formula use semi-perimeter s = (a + b + c)/2, not the full perimeter.
-
Q1 EasyArea of a triangle with base 10 cm and height 6 cm is:ExplanationArea = ½ × base × height = ½ × 10 × 6 = 30 sq cm. Hence the answer is 30 sq cm.
-
Q2 EasyHypotenuse is 13 cm and one leg is 5 cm. The other leg is:ExplanationOther leg = √(13² − 5²) = √(169 − 25) = √144 = 12 cm. Hence the answer is 12 cm.
-
Q3 EasyIn a right triangle, legs are 3 cm and 4 cm. The hypotenuse is:ExplanationBy Pythagoras: c = √(3² + 4²) = √(9 + 16) = √25 = 5 cm. Hence the answer is 5 cm.
-
Q4 EasySides of a triangle are 3 cm, 4 cm and 5 cm. Using Heron’s formula, area is:Explanations = (3 + 4 + 5)/2 = 6. Area = √(6 × 3 × 2 × 1) = √36 = 6 sq cm. Hence the answer is 6 sq cm.
-
Q5 EasyPerimeter of triangle with sides 7, 8 and 9 cm is:ExplanationPerimeter = 7 + 8 + 9 = 24 cm. Hence the answer is 24 cm.
-
Q6 EasyArea with base 14 cm and height 5 cm is:ExplanationArea = ½ × 14 × 5 = 35 sq cm. Hence the answer is 35 sq cm.
-
Q7 MediumAn equilateral triangle has side 6 cm. Its height is:ExplanationHeight = (√3/2) × 6 = 3√3 cm. Hence the answer is 3√3 cm.
-
Q8 MediumIsosceles triangle has equal sides 5 cm and base 6 cm. Area is:ExplanationHalf-base = 3 cm. Height = √(5² − 3²) = 4 cm. Area = ½ × 6 × 4 = 12 sq cm. Hence the answer is 12 sq cm.
-
Q9 MediumA right triangle has legs 8 cm and 15 cm. The hypotenuse is:Explanationc = √(8² + 15²) = √(64 + 225) = √289 = 17 cm. Hence the answer is 17 cm.
-
Q10 MediumIn a 30°-60°-90° triangle, the side opposite 30° is 5 cm. The hypotenuse is:ExplanationSide opposite 30° is half the hypotenuse, so hypotenuse = 2 × 5 = 10 cm. Hence the answer is 10 cm.
-
Q11 MediumTriangle angles in ratio 1 : 2 : 3. It is a:ExplanationAngles: x, 2x, 3x; 6x = 180°, x = 30°. Angles 30°, 60°, 90° — right triangle. Hence the answer is Right triangle.
-
Q12 HardSides 13, 14 and 15 cm. Heron area is:Explanations = 21. Area = √(21 × 8 × 7 × 6) = √7056 = 84 sq cm. Hence the answer is 84 sq cm.
-
Q13 HardBase doubled and height halved. Triangle area:ExplanationOriginal area = ½bh. New = ½(2b)(h/2) = ½bh. Hence the answer is Remains same.
-
Q14 HardEquilateral triangle side 10 cm. Area is:ExplanationArea = (√3/4) × side² = (√3/4) × 100 = 25√3 sq cm. Hence the answer is 25√3 sq cm.
-
DS1 Easy
What is the area of the right triangle?
Statement IThe legs are 6 cm and 8 cm.
Statement IIThe hypotenuse is 10 cm.
ExplanationStatement I alone: area = ½ × 6 × 8 = 24 sq cm. Statement II alone does not fix the legs uniquely. Answer: A. -
DS2 Easy
In a placement drill, what is the hypotenuse of a right triangle?
Statement IOne leg is 9 cm.
Statement IIThe other leg is 12 cm.
ExplanationEach statement alone is insufficient. Together: hypotenuse = √(81 + 144) = 15 cm. Answer: C.
-
DS3 Easy
What is the area of the triangle?
Statement IBase is 12 cm.
Statement IIHeight is 7 cm.
ExplanationNeed both dimensions: area = ½ × 12 × 7 = 42 sq cm. Answer: C. -
DS4 Medium
What is the third side of a triangle with sides 5 cm and 12 cm?
Statement IIt is a right triangle.
Statement IIThe third side is the hypotenuse.
ExplanationTogether we know the right angle is between 5 and 12, so third side = √(25 + 144) = 13 cm. Answer: C.