Mathematics Questions and Answers – Geometry – Section Formula

This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Geometry – Section Formula”.

1. What will be the coordinates of the point which divides the line segment joining the points A(-2, 2) and B(-1, 5) in the ratio 2:5?
a) (\(\frac {-4}{3}, \frac {-20}{9}\))
b) (\(\frac {-4}{3}, \frac {20}{9}\))
c) (\(\frac {4}{3}, \frac {20}{9}\))
d) (\(\frac {4}{3}, \frac {-20}{9}\))
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2. What will be the coordinates of the midpoint of the line segment joining the points (-5, 10) and(15, 2)?
a) (-5, -6)
b) (-5, 6)
c) (5, 6)
d) (5, -6)
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3. In what ratio does the point (\(\frac {-19}{3}, \frac {7}{3}\)) divide the line segment joining A(3, 7) and B(-11, 0)?
a) 1:2 (externally)
b) 1:2 (internally)
c) 2:1 (externally)
d) 2:1 (internally)
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4. What will be the value of y, if the ratio in which the point (\(\frac {3}{4}\), y) divides the line segment joining the points A(-1, 4) and B(6, 5)is 1:3?
a) y = \(\frac {9}{2}\)
b) y = \(\frac {5}{2}\)
c) y = \(\frac {9}{4}\)
d) y = \(\frac {5}{2}\)
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5. What will be ratio in which the line 3x + y – 11 = 0 divides the line segment joining the points (0, -1) and (-3, -4)?
a) 1:2 (internally)
b) 1:2 (externally)
c) 2:1 (externally)
d) 2:1 (internally)
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6. In what ratio is the line segment joining the points A(-5, 2) and B(3, 9) divided by the x-axis?
a) 2:5 (internally)
b) 2:5 (externally)
c) 2:9 (externally)
d) 2:9 (internally)
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7. In what ratio is the line segment joining the points A(2, 4) and B(6, 5) divided by the y-axis?
a) 2:1 (internally)
b) 2:1 (externally)
c) 3:1 (internally)
d) 3:1 (externally)
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8. What will be the coordinates of B, if the point C\((\frac {29}{7}, \frac {46}{7} )\), divides the line segment joining A (5, 8) and B (a, b) in the ratio 2:5?
a) a = 2, b = 3
b) a = -2, b = 3
c) a = 2, b = -3
d) a = -2, b = -3
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9. What will be the length of the median through the vertex A, if the coordinates of the vertices of ∆ABC are A(2, 5), B(5, 0), C(-2, 5)?
a) \(\sqrt {\frac {113}{3}}\) units
b) \(\sqrt {\frac {13}{2}}\) units
c) \(\sqrt {\frac {113}{2}}\) units
d) \(\sqrt {\frac {13}{2}}\) units
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Answer: b
Explanation:

The median through A will bisect the line BC.
Hence, D is the midpoint of BC
Coordinates of D = \((\frac {x_1+x_2}{2}, \frac {y_1+y_2}{2} ) = ( \frac {5-2}{2}, \frac {0-5}{2} ) =( \frac {3}{2}, \frac {-5}{2} )\)
Distance between A and D = \( \sqrt {(x_2-x_1)^2 + (y_2-y_1)^2} \)
= \( \sqrt {(2-\frac {3}{2})^2+ (5+\frac {5}{2})^2} \)
= \( \sqrt {(\frac {1}{2})^2+ (\frac {15}{2})^2} \)
= \( \sqrt {\frac {1}{4}+ \frac {225}{4}} \)
= \( \sqrt {\frac {113}{2}} \) units

10. What will be the coordinates of the fourth vertex S, if P(-1, -1), Q(2, 0), R(2, 3) are the three vertices of a parallelogram?
a) (-5, -12)
b) (5, -12)
c) (5, 12)
d) (-5, 12)
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Answer: c
Explanation:

PQRS is a parallelogram. The opposite side of the parallelogram is equal and parallelogram. Also, the diagonals of the parallelogram bisect each other.
∴ O is the mid-point SQ and PR.
Midpoint of PR
Using, section formula x = \(\frac {mx_2+nx_1}{m+n}\) and y = \(\frac {my_2+ny_1}{m+n}\)
The points are P(-1, -1) and R(2, 3) and the ratio is 1:1
∴ x = \(\frac {1(-1)+1(2)}{2} = \frac {-1+2}{2} = \frac {1}{2}\)
y = \(\frac {1(3)+1(-1)}{2} = \frac {3-1}{2} = \frac {2}{2}\) = 1
Hence, the coordinates of O is (5, 6)
Midpoint of QS
Using, section formula x = \(\frac {mx_2+nx_1}{m+n}\) and y = \(\frac {my_2+ny_1}{m+n}\)
The points are Q(2, 0) and S(a, b) and the ratio is 1:1
∴ x = \(\frac {1(a)+1(2)}{2} = \frac {a+2}{2}\)
y = \(\frac {1(b)+1(0)}{2} = \frac {b}{2}\)
The coordinates of O is (5, 6)
Therefore, \(\frac {a+2}{2}\) = 5
a = 8
\(\frac {b}{2}\) = 6, b = 12
The coordinates of S are (5, 12).

11. What will be the value of a and b, if (-5, a), (-3, -3), (-b, 0) and (-3, 3) are the vertices of the parallelogram?
a) a = 0, b = -1
b) a = -1, b = 1
c) a = 1, b = 1
d) a = 0, b = 1
View Answer

Answer: d
Explanation:

PQRS is a parallelogram. The opposite side of the parallelogram is equal and parallelogram. Also, the diagonals of the parallelogram bisect each other.
∴ O is the mid-point SQ and PR.
Midpoint of PR
Using, section formula x = \(\frac {mx_2+nx_1}{m+n}\) and y = \(\frac {my_2+ny_1}{m+n}\)
The points are P(-5, a) and R(-b, 0) and the ratio is 1:1
∴ x = \(\frac {1(-b)+1(-5)}{2} = \frac {-b-5}{2}\)
y = \(\frac {1(0)+1(a)}{2} = \frac {a}{2}\)
Midpoint of QS
Using, section formula x = \(\frac {mx_2+nx_1}{m+n}\) and y = \(\frac {my_2+ny_1}{m+n}\)
The points are Q(-3, -3) and S(-3, 3) and the ratio is 1:1
∴ x = \(\frac {1(-3)+1(-3)}{2} = \frac {-6}{2}\) = -3
y = \(\frac {1(3)+1(-3)}{2} = \frac {0}{2}\) = 0
Therefore, \(\frac {-b-5}{2}\) = -3
b = 1
\(\frac {a}{2}\) = 0
a = 0

12. What will be the centroid of the ∆ABC whose vertices are A(-2, 4), B(0, 0) and C(4, 2)?
a) (\(\frac {2}{3}\), 2)
b) (\(\frac {2}{3}\), 1)
c) (\(\frac {2}{5}\), 2)
d) (\(\frac {1}{3}\), 2)
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13. The coordinates of one end of the diameter AB of a circle are A (-2, -3) and the coordinates of diameter are (-2, 0). What will be the coordinates of B?
a) (2, -3)
b) (-2, 3)
c) (2, 3)
d) (-2, -3)
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14. The coordinates of the ends of the diameter AB of a circle are A (-4, 7) and B(4, 7). What will be the coordinates of the center of the circle?
a) (0, -8)
b) (0, 8)
c) (0, 7)
d) (0, -7)
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15. The two vertices of ∆ABC are given by A(-3, 0) and B(-8, 5) and its centroid is (-2, 1).What will be the coordinates of the third vertex C?
a) (-5, -2)
b) (5, 2)
c) (-5, 2)
d) (5, -2)
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