Basic Geometric Concepts for Competitive Exam Success

Basic Geometric Concepts - Coordinate Geometry Applications - Case Studies

Q. If two triangles are similar and the lengths of the sides of the first triangle are 3, 4, and 5, what are the lengths of the sides of the second triangle if the shortest side is 6?

A. 6, 8, 10
B. 9, 12, 15
C. 12, 16, 20
D. 15, 20, 25

Solution

The ratio of similarity is 6/3 = 2. Therefore, the sides are 6*2, 4*2, 5*2 = 6, 8, 10.

Correct Answer: A — 6, 8, 10

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Q. If two triangles are similar and the ratio of their corresponding sides is 2:3, what is the ratio of their areas?

A. 4:9
B. 2:3
C. 3:4
D. 1:2

Solution

Area ratio = (2/3)² = 4/9.

Correct Answer: A — 4:9

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Q. What is the circumference of a circle with a diameter of 10?

A. 10π
B. 20π
C. 30π
D. 40π

Solution

Circumference = πd = π(10) = 10π.

Correct Answer: B — 20π

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Q. What is the circumference of a circle with a radius of 7?

A. 14π
B. 21π
C. 28π
D. 35π

Solution

Circumference = 2πr = 2π(7) = 14π.

Correct Answer: B — 21π

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Q. What is the equation of the line that passes through the point (2, 3) with a slope of -1?

A. y = -x + 5
B. y = -x + 3
C. y = x + 1
D. y = -x + 2

Solution

Using point-slope form: y - 3 = -1(x - 2) => y = -x + 5.

Correct Answer: A — y = -x + 5

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

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

Solution

Diagonal = √(8² + 6²) = √(64 + 36) = √100 = 10.

Correct Answer: A — 10

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Q. What is the midpoint of the line segment connecting the points (2, 3) and (4, 7)?

A. (3, 5)
B. (2, 5)
C. (4, 5)
D. (3, 4)

Solution

Midpoint = ((2+4)/2, (3+7)/2) = (3, 5).

Correct Answer: A — (3, 5)

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Q. What is the perimeter of a rectangle with length 4 and width 3?

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

Solution

Perimeter = 2(length + width) = 2(4 + 3) = 2 * 7 = 14.

Correct Answer: B — 12

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Q. What is the perimeter of a triangle with vertices at (0, 0), (0, 3), and (4, 0)?

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

Solution

Sides are 3, 4, and √(3² + 4²) = 5. Perimeter = 3 + 4 + 5 = 12.

Correct Answer: A — 10

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Q. Which of the following points lies inside the circle with center (0, 0) and radius 5?

A. (3, 4)
B. (5, 0)
C. (6, 0)
D. (0, 5)

Solution

Distance from (0,0) to (3,4) is √(3² + 4²) = 5, which is on the circle. (5,0) and (0,5) are on the circle, (6,0) is outside.

Correct Answer: A — (3, 4)

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Q. Which of the following points lies on the line defined by the equation y = 2x + 1?

A. (0, 1)
B. (1, 2)
C. (2, 5)
D. (3, 6)

Solution

For (2, 5): y = 2(2) + 1 = 5, so (2, 5) lies on the line.

Correct Answer: C — (2, 5)

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Q. Which of the following points lies on the line y = 2x + 1?

A. (0, 1)
B. (1, 2)
C. (2, 5)
D. (3, 6)

Solution

Substituting x=2: y = 2(2) + 1 = 5, so (2, 5) lies on the line.

Correct Answer: C — (2, 5)

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Basic Geometric Concepts - Coordinate Geometry Applications - Case Studies

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