Practice Questions on Triangles
Last Updated : 26 Jul, 2024
Triangles are a fundamental concept in geometry, and mastering them is crucial for students at various levels, particularly in class 9. Understanding and solving problems on triangles not only strengthens geometric skills but also enhances overall mathematical reasoning. This collection of practice questions on triangles is designed to cover a wide range of topics, including basic properties, congruence, and classification of triangles.
Whether you are looking for questions on triangles for class 9 or exploring more advanced questions on congruent triangles, this guide provides a set of problems to test and improve your understanding. Dive into practice and perfect your knowledge of triangles with these carefully curated questions.
Types of Triangles
Triangles can be classified based on their sides and angles:
Formulas on Triangle
Some of the common formulas related to triangles are:
| Formula | Description | Expression |
|---|
| Perimeter | Sum of all sides of the triangle | P = a+b+c |
| Area | Area of a triangle using its base and height | Area = (1/2) × b × h |
| Heron's Formula | Area of a triangle using its sides a, b, and c | s = (a+b+c)/2, Area = √[s(s−a)(s−b)(s−c)] |
| Pythagorean Theorem | Relation between the sides of a right triangle | a2 + b2 = c2 |
| Angle Sum Property | Sum of the internal angles of a triangle | Angle A + Angle B + Angle C = 180∘ |
| Similar Triangles | Ratio of corresponding sides of similar triangles | a1/a2 = b1/b2 = c1/c2 |
In this article, we will discuss various problems based on these various concepts.
Practice Questions on Triangles : Solved
1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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 on Triangles : Unsolved
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.
Problem 5: Calculate the area of a triangle with a base of 10 cm and a height of 8 cm.
Problem 6: A triangle has sides measuring 6 cm, 8 cm, and 10 cm. Find the area using Heron's formula.
Problem 7: In a right triangle, the lengths of the legs are 9 cm and 12 cm. Find the length of the hypotenuse.
Problem 8: Triangle ABC is similar to triangle DEF. If the sides of triangle ABC are 3 cm, 4 cm, and 5 cm, and the shortest side of triangle DEF is 6 cm, find the lengths of the other two sides of triangle DEF.
Problem 9: Given a triangle with sides 8 cm and 15 cm, determine the possible range for the third side.
Problem 10: In a triangle with sides 3 cm, 4 cm, and 5 cm, identify the largest angle.
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Conclusion
Mastering triangles is an essential part of geometry education. This collection of problems on triangles and questions on triangles for class 9 offers a thorough practice resource to help you solidify your understanding. By working through these questions on congruent triangles and other triangle-related problems, you can build a strong foundation in geometry. Regular practice will ensure that you are well-prepared for exams and able to apply these concepts effectively in more advanced mathematical studies. Keep practicing and exploring the fascinating world of triangles to achieve excellence in geometry.
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