Mathematics Multiple Choice Questions & Answers 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}))
Answer: b
Clarification: Using, section formula x = (frac {mx_2+nx_1}{m+n}) and y = (frac {my_2+ny_1}{m+n})
The points are A(-2, 2) and B(-1, 5) and the ratio is 2:5
∴ x = (frac {2(-1)+5(-2)}{2+7} = frac {-2-10}{9} = frac {-12}{9} = frac {-4}{3})
y = (frac {2(5)+5(2)}{2+7} = frac {10+10}{9} = frac {20}{9} = frac {20}{9})
Hence, the point is ((frac {-4}{3}, frac {20}{9})).
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)
Answer: c
Clarification: Midpoint lies in the center of the line segment
Hence, it divides the line in the ratio 1:1
Using, section formula x = (frac {mx_2+nx_1}{m+n}) and y = (frac {my_2+ny_1}{m+n})
The points are A(-5, 10) and B(15, 2) and the ratio is 1:1
∴ x = (frac {1(-5)+1(15)}{2} = frac {-5+15}{2} = frac {10}{2}) = 5
y = (frac {1(2)+1(10)}{2} = frac {2+10}{2} = frac {12}{2}) = 6
Hence, the point is (5,6).
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)
Answer: d
Clarification: Let the ratio in which the point ((frac {-19}{3}, frac {7}{3})) divides the line segment joining the points A(3, 7) and B(-11, 0) be k:1
Using, section formula x = (frac {mx_2+nx_1}{m+n}) and y = (frac {my_2+ny_1}{m+n})
The points are A(3, 7) and B(-11, 0) and the ratio is k:1
∴ x = (frac {k(-11)+1(3)}{k+1} = frac {-11k+3}{k+1})
y = (frac {k(0)+1(7)}{k+1} = frac {7}{k+1})
Since, the point is ((frac {-19}{3}, frac {7}{3})).
∴ (frac {-19}{3} = frac {-11k+3}{k+1})
-19(k + 1) = 3(-11k + 3)
-19k – 19 = -33k + 9
-19k + 33k = 19 + 9
14k = 28
k = (frac {28}{14}) = 2
The ratio is 2:1.
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})
Answer: a
Clarification: Using, section formula x = (frac {mx_2+nx_1}{m+n}) and y = (frac {my_2+ny_1}{m+n})
The points are (-1, 4)and B(6, 5)in the ratio 1:3
∴ x = (frac {1(6)+3(-1)}{1+3} = frac {6-3}{4} = frac {3}{4})
y = (frac {1(6)+3(4)}{1+3} = frac {6+12}{4} = frac {18}{4})
Therefore y = (frac {9}{2})
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)
Answer: b
Clarification: Let the ratio in which the line 3x + y – 11 = 0 divides the line segment joining the points (0, -1) and (-3, -4) be k:1.
Using, section formula x = (frac {mx_2+nx_1}{m+n}) and y = (frac {my_2+ny_1}{m+n})
The points are A(0, -1) and B(-3, -4) and the ratio is k:1.
∴ x = (frac {k(-3)+1(0)}{k+1} = frac {-3k}{k+1})
y = (frac {k(-4)+1(-1)}{k+1} = frac {-4k-1}{k+1})
Since, the point ((frac {-3k}{k+1}, frac {-4k-1}{k+1} )) lies on the line 3x+y-11 = 0.
3 ((frac {-3k}{k+1} + frac {-4k-1}{k+1} ))-11 = 0
3(-3k) + (-4k – 1) – 11(k + 1) = 0
-9k – 4k – 1 – 11k – 11 = 0
-24k – 12 = 0
-24k = 12
k = (frac {12}{-24} = frac {-1}{2})
The ratio is 1:2 (externally).
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)
Answer: c
Clarification: Let the ratio in which the x-axis divides the line segment joining the points A(-5, 2) and B(3, 9) be k:1
Using, section formula x = (frac {mx_2+nx_1}{m+n}) and y = (frac {my_2+ny_1}{m+n})
The points are A(-5, 2) and B(3, 9) and the ratio is k:1
∴ x = (frac {k(3)+1(-5)}{k+1} = frac {3k-5}{k+1})
y = (frac {k(9)+1(2)}{k+1} = frac {9k+2}{k+1})
Since, the point is on x-axis.
Hence, the y-coordinate will be zero.
∴ 0 = (frac {9k+2}{k+1})
0 = 9k+2
k = (frac {-2}{9})
The ratio in which the y-axis cuts the line segment joining the points A(-5, 2) and B(3, 9) will be 2:9 (externally).
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)
Answer: d
Clarification: Let the ratio in which the y-axis divides the line segment joining the points A(2, 4) and B(6, 5) be k:1
Using, section formula x = (frac {mx_2+nx_1}{m+n}) and y = (frac {my_2+ny_1}{m+n})
The points are A(2, 4) and B(6, 5) and the ratio is k:1
∴ x = (frac {k(6)+1(2)}{k+1} = frac {6k+2}{k+1})
y = (frac {k(5)+1(4)}{k+1} = frac {5k+4}{k+1})
Since, the point is on y-axis.
Hence, the x-coordinate will be zero.
∴ 0 = (frac {6k+2}{k+1})
0 = 6k + 2
k = (frac {-6}{2}) = -3
The ratio in which the y-axis cuts the line segment joining the points A(2, 4) and B(6, 5) will be 3:1 (externally).
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
Answer: a
Clarification: Using, section formula x = (frac {mx_2+nx_1}{m+n}) and y = (frac {my_2+ny_1}{m+n})
The points are A(5, 8)and B(a, b)in the ratio 2:5
∴ x = (frac {2(a)+5(5)}{2+5} = frac {2a+25}{7})
y = (frac {2(b)+5(8)}{2+5} = frac {2b+40}{7})
But the coordinates of C are ((frac {29}{7}, frac {46}{7} ))
Therefore, (frac {2a+25}{7} = frac {29}{7})
a = 2
(frac {2b+40}{7} = frac {46}{7})
b = 3
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
Answer: b
Clarification:
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)
Answer: c
Clarification:
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).
