A triangle is a closed two-dimensional geometric shape that is formed by joining three line segments. It has three sides, three vertices (corners), and three angles.
Solved Questions on Triangles
Q1. Classify the triangle with sides of lengths 7 cm, 24 cm, and 25 cm.
To classify the triangle, we can use the Pythagorean theorem to determine if it is a right triangle.
According to the theorem, in a right triangle, the square of the length of the hypotenuse (the longest side) should be equal to the sum of the squares of the other two sides.
252 = 72 + 242 625 = 49 + 576 625 = 625
Since the equation holds true, the triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle. Additionally, since it has all different side lengths, it is also a scalene triangle.
Q2. Find the area of a triangle with a base of 10 cm and a height of 5 cm.
The area (A) of a triangle is given by the formula:
Area = (1/2) × base × height
Substituting the given values:
Area = (1/2) × 10 × 5 Area = (1/2) × 50 Area = 25 square centimeters
So, the area of the triangle is 25 square centimeters.
Q3. Calculate the perimeter of an equilateral triangle with each side measuring 8 cm.
The perimeter (P) of an equilateral triangle is the sum of the lengths of all its sides. Since all sides of an equilateral triangle are equal:
Perimeter = 3 × side length
Perimeter = 3 × 8 Perimeter = 24 cm
So, the perimeter of the equilateral triangle is 24 centimeters.
Q4. Can a triangle have sides of lengths 3 cm, 4 cm, and 8 cm?
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let's check this for the given sides:
- 3 + 4 > 8 (7 > 8, which is false)
- 3 + 8 > 4 (11 > 4, which is true)
- 4 + 8 > 3 (12 > 3, which is true)
Since the first condition fails, a triangle with sides of lengths 3 cm, 4 cm, and 8 cm cannot exist.
Q5. Two triangles are similar. The sides of the first triangle are 6 cm, 8 cm, and 10 cm. The shortest side of the second triangle is 3 cm. Find the lengths of the other two sides of the second triangle.
Since the triangles are similar, the corresponding sides are proportional. The ratio of the sides of the first triangle to the second triangle is the same.
The shortest side of the first triangle is 6 cm, and the shortest side of the second triangle is 3 cm. The ratio of the sides is:
Ratio = 3/6 = 1/2
Using this ratio, we can find the other sides of the second triangle:
For the side corresponding to 8 cm: Other side = 8 × (1/2) = 4 cm
For the side corresponding to 10 cm: Other side = 10 × (1/2) = 5 cm
So, the lengths of the other two sides of the second triangle are 4 cm and 5 cm.
Q6. In a right triangle, one leg is 9 cm and the hypotenuse is 15 cm. Find the length of the other leg.
Let the length of the other leg be (b). According to the Pythagorean theorem:
a2 + b2 = c2
Here, a = 9 cm and c = 15 cm. Substituting these values in:
92 + b2 = 152
⇒ 81 + b2 = 225
⇒ b2 = 225 - 81
⇒ b2 = 144
⇒ b = √144
⇒ b = 12 cm
So, the length of the other leg is 12 cm.
Q7. Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.
First, calculate the semi-perimeter (s) of the triangle:
s = (a + b + c) / 2 s = (7 + 8 + 9) / 2 s = 12 cm
Using Heron's formula, the area (A) of the triangle is:
A = √[s(s-a)(s-b)(s-c)]
Substitute the side lengths into the formula:
A = √[12(12-7)(12-8)(12-9)] A = √[12 × 5 × 4 × 3] A = √720 A ≈ 26.83 square centimeters
So, the area of the triangle is approximately 26.83 square centimeters.
Q8. In a triangle, one angle is 35 degrees and another angle is 65 degrees. Find the measure of the third angle.
The sum of the angles in any triangle is always 180 degrees. Let the third angle be (x). Then:
35 + 65 + x = 180 100 + x = 180 x = 180 - 100 x = 80
So, the measure of the third angle is 80 degrees.
Practice Questions
Following are some practice questions triangle which you must try have better command over the topic.
Problem 1: Given a triangle with sides 5 cm, 5 cm, and 8 cm, classify the triangle based on its sides.
Problem 2: In a triangle, two angles are 45° and 55°. Find the measure of the third angle.
Problem 3: In a triangle, one angle measures 80° and another angle measures 60°. What is the measure of the third angle?
Problem 4: Find the perimeter of a triangle with sides measuring 7 cm, 10 cm, and 12 cm.
