Triangles - Properties & Pythagoras for Competitive Exams

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Triangles - Properties & Pythagoras

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Triangles - Properties & Pythagoras MCQ & Objective Questions

Understanding "Triangles - Properties & Pythagoras" is crucial for students preparing for school exams and competitive assessments. This topic not only forms the foundation of geometry but also plays a significant role in various objective questions and MCQs. By practicing these important questions, students can enhance their problem-solving skills and boost their confidence for exam day.

What You Will Practise Here

Types of triangles: scalene, isosceles, and equilateral
Properties of triangles: angles, sides, and their relationships
Pythagorean theorem: understanding and applying the formula
Area and perimeter calculations for different triangles
Congruence and similarity in triangles
Real-life applications of triangles in various fields
Diagrams and visual representations to aid understanding

Exam Relevance

The topic of triangles is frequently tested in CBSE, State Boards, NEET, and JEE exams. Students can expect questions that require them to apply the Pythagorean theorem, identify properties of different triangles, and solve problems involving area and perimeter. Common question patterns include multiple-choice questions that assess both conceptual understanding and practical application of theorems.

Common Mistakes Students Make

Confusing the properties of different types of triangles
Misapplying the Pythagorean theorem in non-right triangles
Overlooking the importance of units when calculating area and perimeter
Failing to visualize problems with diagrams, leading to misunderstandings

FAQs

Question: What is the Pythagorean theorem?
Answer: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Question: How do I find the area of a triangle?
Answer: The area of a triangle can be calculated using the formula: Area = 1/2 × base × height.

Now is the time to strengthen your understanding of triangles! Dive into our practice MCQs and test your knowledge on "Triangles - Properties & Pythagoras". With consistent practice, you will be well-prepared to tackle any exam questions confidently.

Q. A right triangle has an area of 30 square units and one leg measuring 10 units. What is the length of the other leg?

A. 3
B. 6
C. 9
D. 12

Solution

Area = (1/2) * base * height, so 30 = (1/2) * 10 * height. Thus, height = 6.

Correct Answer: B — 6

Q. A right triangle has an area of 30 square units and one leg measuring 5 units. What is the length of the other leg?

A. 6
B. 10
C. 12
D. 15

Solution

Area = (1/2) * base * height, so 30 = (1/2) * 5 * height. Thus, height = 30 * 2 / 5 = 12.

Correct Answer: B — 10

Q. A right triangle has one angle measuring 30 degrees. If the hypotenuse is 10, what is the length of the side opposite the 30-degree angle?

A. 5
B. 6
C. 7
D. 8

Solution

The side opposite the 30-degree angle is half the hypotenuse: 10 * 1/2 = 5.

Correct Answer: A — 5

Q. A right triangle has one leg measuring 9 and the hypotenuse measuring 15. What is the length of the other leg?

A. 12
B. 10
C. 8
D. 6

Solution

Using the Pythagorean theorem, b = √(c² - a²) = √(15² - 9²) = √(225 - 81) = √144 = 12.

Correct Answer: A — 12

Q. A right triangle has one leg of length 9 and a hypotenuse of length 15. What is the length of the other leg?

A. 12
B. 10
C. 8
D. 6

Solution

Using the Pythagorean theorem, b = √(c² - a²) = √(15² - 9²) = √(225 - 81) = √144 = 12.

Correct Answer: A — 12

Q. A triangle has angles measuring 30°, 60°, and 90°. If the shortest side is 5, what is the length of the hypotenuse?

A. 5
B. 10
C. 7.5
D. 8.66

Solution

In a 30-60-90 triangle, the hypotenuse is twice the shortest side. Therefore, hypotenuse = 2 * 5 = 10.

Correct Answer: B — 10

Q. A triangle has sides of lengths 7, 24, and 25. Is this triangle a right triangle?

A. Yes
B. No
C. Cannot be determined
D. Only if angles are known

Solution

Using the Pythagorean theorem, check if 25² = 7² + 24². 625 = 49 + 576, which is true. Thus, it is a right triangle.

Correct Answer: A — Yes

Q. If a right triangle has a hypotenuse of 10 and one leg of 6, what is the length of the other leg?

A. 8
B. 7
C. 5
D. 4

Solution

Using the Pythagorean theorem, b = √(c² - a²) = √(10² - 6²) = √(100 - 36) = √64 = 8.

Correct Answer: A — 8

Q. If a right triangle has an angle of 30 degrees, what is the ratio of the lengths of the opposite side to the hypotenuse?

A. 1:2
B. 1:√3
C. √3:1
D. 2:1

Solution

In a 30-60-90 triangle, the ratio of the opposite side to the hypotenuse is 1:2.

Correct Answer: A — 1:2

Q. If a triangle has sides of lengths 7, 24, and 25, is it a right triangle?

A. Yes
B. No
C. It depends
D. Cannot be determined

Solution

Check using the Pythagorean theorem: 25² = 7² + 24², 625 = 49 + 576, 625 = 625. It is a right triangle.

Correct Answer: A — Yes

Q. If the lengths of the legs of a right triangle are 12 and 16, what is the length of the hypotenuse?

A. 20
B. 22
C. 24
D. 26

Solution

Using the Pythagorean theorem, c = √(12² + 16²) = √(144 + 256) = √400 = 20.

Correct Answer: A — 20

Q. If the lengths of the legs of a right triangle are 8 and 15, what is the length of the hypotenuse?

A. 17
B. 18
C. 19
D. 20

Solution

Using the Pythagorean theorem, c = √(8² + 15²) = √(64 + 225) = √289 = 17.

Correct Answer: A — 17

Q. If the lengths of the legs of a right triangle are 9 and 12, what is the perimeter of the triangle?

A. 30
B. 32
C. 34
D. 36

Solution

Hypotenuse = √(9² + 12²) = √(81 + 144) = √225 = 15. Perimeter = 9 + 12 + 15 = 36.

Correct Answer: B — 32

Q. If the lengths of the legs of a right triangle are equal, what type of triangle is it?

A. Isosceles
B. Equilateral
C. Scalene
D. Right

Solution

A right triangle with equal legs is an isosceles triangle.

Correct Answer: A — Isosceles

Q. If the lengths of the sides of a triangle are 5, 12, and 13, what type of triangle is it?

A. Acute
B. Obtuse
C. Right
D. Equilateral

Solution

Since 5² + 12² = 13², it is a right triangle.

Correct Answer: C — Right

Q. If the lengths of the sides of a triangle are 9, 12, and 15, is it a right triangle?

A. Yes
B. No
C. It depends
D. Cannot be determined

Solution

Check using the Pythagorean theorem: 15² = 9² + 12², 225 = 81 + 144, 225 = 225. It is a right triangle.

Correct Answer: A — Yes

Q. If the lengths of the two legs of a right triangle are equal, what is the measure of the angles opposite those legs?

A. 45 degrees
B. 60 degrees
C. 30 degrees
D. 90 degrees

Solution

In an isosceles right triangle, the angles opposite the equal legs are both 45 degrees.

Correct Answer: A — 45 degrees

Q. In a right triangle, if one leg is 5 and the hypotenuse is 13, what is the length of the other leg?

A. 12
B. 10
C. 8
D. 6

Solution

Using the Pythagorean theorem, b = √(c² - a²) = √(13² - 5²) = √(169 - 25) = √144 = 12.

Correct Answer: A — 12

Q. In a right triangle, if the angles are 30° and 60°, what is the ratio of the lengths of the sides opposite these angles?

A. 1:√3
B. 1:2
C. 1:1
D. 1:√2

Solution

The sides opposite 30° and 60° are in the ratio 1:√3.

Correct Answer: A — 1:√3

Q. In a right triangle, if the hypotenuse is 10 and one leg is 6, what is the length of the other leg?

A. 4
B. 5
C. 6
D. 8

Solution

Using the Pythagorean theorem, b = √(c² - a²) = √(10² - 6²) = √(100 - 36) = √64 = 8.

Correct Answer: D — 8

Q. In a right triangle, if the lengths of the legs are 8 and 15, what is the area of the triangle?

A. 60
B. 80
C. 120
D. 100

Solution

Area = (1/2) * base * height = (1/2) * 8 * 15 = 60.

Correct Answer: A — 60

Q. In a right triangle, if the lengths of the legs are 8 and 15, what is the length of the hypotenuse?

A. 17
B. 18
C. 19
D. 20

Solution

Using the Pythagorean theorem, c = √(8² + 15²) = √(64 + 225) = √289 = 17.

Correct Answer: A — 17

Q. In a right triangle, if the lengths of the legs are equal, what is the measure of the angles opposite those legs?

A. 45 degrees
B. 60 degrees
C. 30 degrees
D. 90 degrees

Solution

In an isosceles right triangle, the angles opposite the equal legs are both 45 degrees.

Correct Answer: A — 45 degrees

Q. In a right triangle, if the lengths of the legs are in the ratio 3:4, what is the length of the hypotenuse if the shorter leg is 9?

A. 12
B. 15
C. 18
D. 21

Solution

If the shorter leg is 9, the longer leg is (4/3) * 9 = 12. Hypotenuse = √(9² + 12²) = √(81 + 144) = √225 = 15.

Correct Answer: B — 15

Q. What is the area of a right triangle with legs measuring 6 and 8?

A. 24
B. 30
C. 48
D. 36

Solution

Area = (1/2) * base * height = (1/2) * 6 * 8 = 24.

Correct Answer: A — 24

Q. What is the length of the diagonal of a rectangle with sides of lengths 6 and 8?

A. 10
B. 12
C. 14
D. 16

Solution

Using the Pythagorean theorem, diagonal = √(6² + 8²) = √(36 + 64) = √100 = 10.

Correct Answer: A — 10

Q. What is the length of the hypotenuse of a right triangle with legs of lengths 3 and 4?

A. 5
B. 6
C. 7
D. 8

Solution

Using the Pythagorean theorem, c = √(a² + b²) = √(3² + 4²) = √(9 + 16) = √25 = 5.

Correct Answer: A — 5

Q. What is the perimeter of a right triangle with legs of lengths 5 and 12?

A. 30
B. 25
C. 20
D. 22

Solution

First, find the hypotenuse: c = √(5² + 12²) = √(25 + 144) = √169 = 13. Perimeter = 5 + 12 + 13 = 30.

Correct Answer: A — 30

Q. What is the perimeter of a right triangle with legs of lengths 9 and 12?

A. 30
B. 32
C. 34
D. 36

Solution

Hypotenuse = √(9² + 12²) = √(81 + 144) = √225 = 15. Perimeter = 9 + 12 + 15 = 36.

Correct Answer: B — 32

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