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Midpoint formula and Solutions

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Question 11

Question: In what ratio does the y-axis divide the line AB, Where A(-4, 1) and B(17, 10). [2 Mark]

Answer: Let P divide the (1995 - year) given line in the ratio m:n.

Question 12

Question: Find the co-ordinates of the point which divides the line segment joining the given points in the given ratio internally (-4, 1) and (17, 10); 1:2. [2 Mark]

Answer: Let A(-4, 1), B(17, 10) and ratio m:n = 1:2

Co-ordinates of point dividing the line segment,

= (3, 4)

Question 13

Question: The mid point of the line segment AB shown in the diagram, is (4, -3). Write down the co-ordinates of A and B. [2 Mark]

Answer:

The co-ordinates of A (x₁, 0), B(0, y₂)

(points are A on x-axis, B on y-axis)

Substituting,

Hence co-ordinates of A(8, 0), B(0, -6).

Question 14

Question: Calculate the distance between A(7, 3) and B on the x-axis whose abscissa is 11. [2 Mark]

Answer: Point on x-axis. Hence its ordinate = 0.

Co-ordinates of B(11, 0), A(7, 3).

By distance formula, distance between AB,

= 5 units

Question 15

Question: The centre O, of a circle has the co-ordinates (4, 5) and one point on the circumference is (8, 10), find the co-ordinates of the other end of the diameter of the circle through this point. [2 Mark]

Answer: Let AB be the diameter.

O centre lies on diameter.

A(8, 10), O(4, 5), B(a, b)

O is the centre mid point of diameter.

10 = 10 + b

Hence co-ordinates of B(0, 0) [B the other end of the diameter]

Question 16

Question: Calculate the ratio in which the line joining A(6, 5) and B(4, -3) is divided by the line y = 2. [3 Mark]

Answer: Let the line y = 2(parallel to x-axis) divide line AB at O.

Coordinates of O(x, 2) in the ratio m: n.

Then according to section formula,

Question 17

Question: Find the area of a triangle whose vertices are (-5, -1), (3, -5) and (5, 2). [2 Mark]

Answer: Area of triangle

= 26

= 26 Sq. Units.

Question 18

Question: Find the area of the triangle formed by joining the mid points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle. [3 Mark]

Answer:

E is the mid point of AB. the coordinates of E are = = (1, 0)

D is the mid point of BC. the coordinates of D are = = (1, 2)

F is the mid point of CA. the coordinates of F are = = (0, 1)

Area of DEF =

=

Area of ABC =

= 4 Sq. units

Ratio of the area of DEF to the area of ABC is.

Area of DEF = 1 Sq. units, Area of ABC is 4 Sq. units. Ratio is .

Question 19

Question: Find the area of the quadrilateral whose vertices, taken in order are (-4, -2), (-3, -5), (3, -2) and (2, 3). [3 Mark]

Answer:

Given A (-4, -2), B (-3, -5), C (3, -2), D (2, 3) be the 4 vertices of a quadrilateral. Join BD.

Area of quadrilateral ABCD is equal to sum of the area of the triangle ABD and DBC.

A (-4, -2), B (-3, -5),D (2, 3)

Area of ABD =

= Sq. units.

D (2, 3), B (-3, -5), C (3, -2)

Area of DBC =

= Sq. units

Area of the quadrilateral ABCD = = 28 Squints