Coordinate Geometry - Distance & Section Formula for Exams

Coordinate Geometry - Distance and Section Formula - Problems on Triangles - Applications

Q. Find the coordinates of the point that divides the segment joining (2, 3) and (4, 7) in the ratio 1:3.

A. (3, 5)
B. (2.5, 4)
C. (3.5, 5.5)
D. (3, 6)

Solution

Using the section formula: P(x, y) = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)), where m=1, n=3, x1=2, y1=3, x2=4, y2=7. P = ((1*4 + 3*2)/(1+3), (1*7 + 3*3)/(1+3)) = (3, 5).

Correct Answer: A — (3, 5)

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Q. If a triangle has vertices at (1, 1), (4, 1), and (1, 5), what is its perimeter?

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

Solution

Perimeter = AB + BC + CA = 3 + 3 + 4 = 10.

Correct Answer: B — 10

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Q. If a triangle has vertices at (1, 2), (4, 6), and (7, 2), what is its perimeter?

A. 18
B. 20
C. 22
D. 16

Solution

Calculate the lengths of the sides using the distance formula: AB = √((4-1)² + (6-2)²) = 5, BC = √((7-4)² + (2-6)²) = 5, CA = √((1-7)² + (2-2)²) = 6. Perimeter = 5 + 5 + 6 = 16.

Correct Answer: A — 18

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Q. If point P divides the segment joining A(1, 2) and B(5, 6) in the ratio 1:3, what are the coordinates of P?

A. (3, 4)
B. (2, 3)
C. (4, 5)
D. (1, 2)

Solution

Using the section formula: P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) = ((1*5 + 3*1)/(1+3), (1*6 + 3*2)/(1+3)) = (3, 4).

Correct Answer: A — (3, 4)

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Q. If the coordinates of the vertices of a triangle are (1, 1), (4, 1), and (1, 5), what is the length of the base?

A. 3
B. 4
C. 5
D. 6

Solution

The base is the distance between (1, 1) and (4, 1): d = √((4-1)² + (1-1)²) = √9 = 3.

Correct Answer: A — 3

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Q. What is the area of a triangle with vertices at (0, 0), (b, 0), and (0, h)?

A. bh/2
B. bh
C. b+h
D. b*h/2

Solution

Area = 1/2 * base * height = 1/2 * b * h = bh/2.

Correct Answer: A — bh/2

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

A. 1
B. 2
C. 3
D. 0

Solution

Distance = |Ax + By + C| / √(A² + B²) = |2(2) + 3(3) - 6| / √(2² + 3²) = |4 + 9 - 6| / √13 = 7/√13.

Correct Answer: A — 1

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Q. What is the distance from the point (3, 4) to the line defined by the equation 2x + 3y - 12 = 0?

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

Solution

Using the formula for distance from a point to a line: d = |Ax + By + C| / √(A² + B²), where A=2, B=3, C=-12. d = |2*3 + 3*4 - 12| / √(2² + 3²) = |6 + 12 - 12| / √13 = 6/√13.

Correct Answer: A — 2

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Q. What is the length of the line segment connecting the points (1, 2) and (1, 5)?

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

Solution

Since the x-coordinates are the same, the distance is the difference in y-coordinates: d = |y2 - y1| = |5 - 2| = 3.

Correct Answer: A — 3

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Coordinate Geometry - Distance and Section Formula - Problems on Triangles - Applications

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