Coordinate Geometry - Distance & Section Formula for Exams

Coordinate Geometry - Distance and Section Formula - Problems on Triangles - Problem Set

Q. If point C(0, 0) and point D(6, 8) are the endpoints of a line segment, what is the length of CD?

A. 8.0
B. 10.0
C. 6.0
D. 7.0

Solution

Using the distance formula: d = √((6 - 0)² + (8 - 0)²) = √(36 + 64) = √100 = 10.0.

Correct Answer: B — 10.0

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Q. If point D divides the segment AB with A(2, 3) and B(8, 7) in the ratio 2:3, what are the coordinates of D?

A. (4.8, 5.2)
B. (5.2, 4.8)
C. (6, 5)
D. (5, 5)

Solution

Using the section formula: D = ((2*8 + 3*2)/(2+3), (2*7 + 3*3)/(2+3)) = (4.8, 5.2).

Correct Answer: A — (4.8, 5.2)

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Q. If point E(1, 1) and point F(4, 5) are the endpoints of a line segment, what is the length of EF?

A. 3.61
B. 4.24
C. 5.0
D. 6.0

Solution

Using the distance formula: d = √((4 - 1)² + (5 - 1)²) = √(9 + 16) = √25 = 5.0.

Correct Answer: B — 4.24

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Q. If triangle ABC has vertices A(1, 2), B(4, 6), and C(7, 2), what is the area of triangle ABC?

A. 12
B. 10
C. 14
D. 8

Solution

Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| = 0.5 * |1(6 - 2) + 4(2 - 2) + 7(2 - 6)| = 0.5 * |4 + 0 - 28| = 12.

Correct Answer: B — 10

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Q. If triangle ABC has vertices A(1, 2), B(4, 6), and C(7, 2), what is the length of side AB?

A. 5.0
B. 4.24
C. 3.0
D. 6.0

Solution

Using the distance formula: AB = √((4 - 1)² + (6 - 2)²) = √(9 + 16) = √25 = 5.0.

Correct Answer: B — 4.24

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Q. What is the area of triangle formed by points A(0, 0), B(4, 0), and C(0, 3)?

A. 6.0
B. 12.0
C. 8.0
D. 10.0

Solution

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.0.

Correct Answer: A — 6.0

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Q. What is the distance between the points (5, 5) and (5, 10)?

A. 5.0
B. 10.0
C. 15.0
D. 0.0

Solution

Using the distance formula: d = √((5 - 5)² + (10 - 5)²) = √(0 + 25) = √25 = 5.0.

Correct Answer: A — 5.0

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Q. What is the distance from the point (3, 4) to the line defined by the equation 2x + 3y - 6 = 0?

A. 2.0
B. 1.0
C. 3.0
D. 4.0

Solution

Distance = |Ax1 + By1 + C| / √(A² + B²) = |2*3 + 3*4 - 6| / √(2² + 3²) = |6 + 12 - 6| / √13 = 12/√13 ≈ 1.0.

Correct Answer: B — 1.0

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Q. What is the length of the altitude from point C(7, 2) to side AB of triangle ABC with A(1, 2) and B(4, 6)?

A. 3.0
B. 2.0
C. 4.0
D. 5.0

Solution

Using the formula for the area of a triangle: Area = 1/2 * base * height. Area = 1/2 * 5 * height. Area = 5.0. Height = 3.0.

Correct Answer: A — 3.0

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Q. What is the length of the altitude from vertex A(1, 1) to the base BC of triangle ABC with B(4, 1) and C(4, 5)?

A. 3
B. 4
C. 2
D. 5

Solution

The base BC is vertical, so the length of the altitude is the horizontal distance from A to line x = 4, which is |4 - 1| = 3.

Correct Answer: A — 3

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Q. What is the midpoint of the line segment joining points (4, 2) and (10, 6)?

A. (7, 4)
B. (6, 4)
C. (5, 4)
D. (8, 3)

Solution

Midpoint M = ((4 + 10)/2, (2 + 6)/2) = (7, 4).

Correct Answer: A — (7, 4)

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Q. What is the section formula for dividing the line segment joining points (2, 3) and (8, 7) in the ratio 1:3?

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

Solution

Using the section formula: P(x, y) = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) where m=1, n=3. P = ((1*8 + 3*2)/(1+3), (1*7 + 3*3)/(1+3)) = (6, 5).

Correct Answer: B — (6, 5)

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Coordinate Geometry - Distance and Section Formula - Problems on Triangles - Problem Set

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