Question 1

A point $P(x, y)$ divides the line segment joining $A(3, 6)$ and $B(-3, 4)$ in the ratio 2:3. What is the value of $x$?

Accepted Answer

Correct Option: A. Solution: Using the section formula $x = \frac{2(-3) + 3(3)}{2 + 3} = 0$.

Question 2

A point $P(x, y)$ divides the line segment joining $A(2, 3)$ and $B(6, 7)$ in the ratio $3:1$. What are the coordinates of $P$?

Accepted Answer

Correct Option: B. Solution: Using the section formula, $x = \frac{3 	imes 6 + 1 	imes 2}{3 + 1} = 4$ and $y = \frac{3 	imes 7 + 1 	imes 3}{3 + 1} = 5$. Thus, $P(4, 5)$.

Question 3

If a point $P$ divides the line segment joining $A(2, -2)$ and $B(-7, 4)$ in the ratio 1:2, what are the coordinates of $P$?

Accepted Answer

Correct Option: A. Solution: Using the section formula, the coordinates of $P$ are $\left( \frac{1(-7) + 2(2)}{1+2}, \frac{1(4) + 2(-2)}{1+2} \right) = (-1, 0)$.

Question 4

What is the length of the line segment joining the points $(-3, 4)$ and $(3, -4)$?

Accepted Answer

Correct Option: B. Solution: Using the distance formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, the distance is $\sqrt{(3 - (-3))^2 + (-4 - 4)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.

Question 5

If a point $P(x, y)$ divides the line segment joining $A(-3, 4)$ and $B(5, -2)$ externally in the ratio $2:3$, what is the x-coordinate of $P$?

Accepted Answer

Correct Option: B. Solution: For external division, $x = \frac{2 	imes 5 - 3 	imes (-3)}{2 - 3} = \frac{10 + 9}{-1} = \frac{19}{-1} = -19$. The correct answer is $\frac{11}{5}$, as the calculation should be adjusted for external division.

Question 6

Find the coordinates of a point on the y-axis which is equidistant from the points $(6, 5)$ and $(-4, 3)$.

Accepted Answer

Correct Option: C. Solution: The point on the y-axis equidistant from $(6, 5)$ and $(-4, 3)$ is $(0, 9)$.

Question 7

Given points $A(1, 5)$, $B(2, 3)$, and $C(-2, -11)$, determine if they are collinear.

Accepted Answer

Correct Option: A. Solution: To check collinearity, calculate the distances $AB$, $BC$, and $AC$. If the sum of two distances equals the third, the points are collinear.

Question 8

What is the formula to calculate the distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ in a Cartesian plane?

Accepted Answer

Correct Option: A. Solution: The distance formula is derived from the Pythagorean theorem and is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Question 9

Given the points $A(2, 3)$ and $B(10, 7)$, find the coordinates of the point $P(x, y)$ that divides the line segment $AB$ internally in the ratio $3:2$.

Accepted Answer

Correct Option: A. Solution: Using the section formula: $x = \frac{3 	imes 10 + 2 	imes 2}{3 + 2} = 6$, $y = \frac{3 	imes 7 + 2 	imes 3}{3 + 2} = 5$. Thus, $P(6, 5)$.

Question 10

A point $Q$ divides the line segment joining $A(2, -2)$ and $B(-7, 4)$ in the ratio 2:1. What are the coordinates of $Q$?

Accepted Answer

Correct Option: A. Solution: Using the section formula, the coordinates of $Q$ are $\left( \frac{2(-7) + 1(2)}{2+1}, \frac{2(4) + 1(-2)}{2+1} \right) = (-4, 2)$.

Question 11

Which of the following points is the midpoint of the line segment joining $A(4, -3)$ and $B(8, 5)$?

Accepted Answer

Correct Option: A. Solution: The midpoint of a line segment is given by $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$. For $A(4, -3)$ and $B(8, 5)$, the midpoint is $(6, 1)$.

Question 12

In a coordinate plane, if the distance between the origin $(0, 0)$ and a point $(x, y)$ is $10$ units, what could be the possible coordinates of the point?

Accepted Answer

Correct Option: A. Solution: The distance from the origin to $(x, y)$ is given by $\sqrt{x^2 + y^2} = 10$. Solving $x^2 + y^2 = 100$ for integer solutions, we find $(6, 8)$ and $(8, 6)$ are valid solutions.

Question 13

Consider two points $A(4, 2)$ and $B(8, 10)$. Find the equation of the line such that any point $(x, y)$ on this line is equidistant from both $A$ and $B$.

Accepted Answer

Correct Option: B. Solution: The perpendicular bisector of the segment $AB$ will have the equation of the form $x - y = c$. Calculating the midpoint of $AB$, we have $\left(\frac{4+8}{2}, \frac{2+10}{2}\right) = (6, 6)$. Using the midpoint in the equation form $x - y = c$, we find $c = 6$. Thus, the equation is $x - y = 6$.

Question 14

The midpoint of a line segment is $(3, 4)$. If one endpoint is $(1, 2)$, what is the other endpoint?

Accepted Answer

Correct Option: A. Solution: Let the other endpoint be $(x, y)$. Using the midpoint formula, $$\left( \frac{1+x}{2}, \frac{2+y}{2} \right) = (3, 4)$$. Solving, we get $x = 5$ and $y = 6$. Thus, the other endpoint is $(5, 6)$.

Question 15

A line segment is divided by a point $P$ such that the ratio of the segments is $1:1$. If the endpoints of the line segment are $C(4, -2)$ and $D(10, 6)$, what are the coordinates of point $P$?

Accepted Answer

Correct Option: A. Solution: Since the line segment is divided in the ratio $1:1$, point $P$ is the midpoint. Using the midpoint formula, $$\left( \frac{4+10}{2}, \frac{-2+6}{2} \right) = (7, 2)$$.

Question 16

What is the formula to find the coordinates of a point $P(x, y)$ that divides a line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m_1:m_2$?

Accepted Answer

Correct Option: A. Solution: The section formula for internal division is given by $x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}$ and $y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2}$.

Question 17

If a point $P(x, y)$ is equidistant from points $A(1, 2)$ and $B(4, 6)$, what is the equation relating $x$ and $y$?

Accepted Answer

Correct Option: D. Solution: For $P(x, y)$ to be equidistant from $A(1, 2)$ and $B(4, 6)$, $AP = BP$. Thus, $(x - 1)^2 + (y - 2)^2 = (x - 4)^2 + (y - 6)^2$. Simplifying gives $x - y = 2$.

Question 18

If a point $Q(x, y)$ is equidistant from points $A(7, 1)$ and $B(3, 5)$, what is the relationship between $x$ and $y$?

Accepted Answer

Correct Option: A. Solution: Since $Q$ is equidistant from $A$ and $B$, $AP = BP$. Solving $\sqrt{(x - 7)^2 + (y - 1)^2} = \sqrt{(x - 3)^2 + (y - 5)^2}$ gives $x - y = 2$.

Question 19

In a parallelogram, the diagonals bisect each other. Given the vertices $A(-3, 0)$, $B(3, 0)$, $C(2, 5)$, and $D(-4, 5)$, calculate the coordinates of the intersection point of the diagonals.

Accepted Answer

Correct Option: A. Solution: The intersection point of the diagonals of a parallelogram is the midpoint of both diagonals. For diagonal $AC$, the midpoint is $\left(\frac{-3+2}{2}, \frac{0+5}{2}\right) = (-0.5, 2.5)$. For diagonal $BD$, the midpoint is $\left(\frac{3+(-4)}{2}, \frac{0+5}{2}\right) = (-0.5, 2.5)$. Thus, the intersection point is $(0, 2.5)$.

Question 20

A point $P(0, y)$ on the y-axis is equidistant from the points $A(3, 4)$ and $B(3, -4)$. What is the value of $y$?

Accepted Answer

Correct Option: A. Solution: The point $P$ is equidistant from $A$ and $B$, so $\sqrt{(0-3)^2 + (y-4)^2} = \sqrt{(0-3)^2 + (y+4)^2}$. Solving gives $y = 0$.

Question 21

What is the distance between the points $P(4, 6)$ and $Q(6, 8)$ using the distance formula?

Accepted Answer

Correct Option: A. Solution: Using the distance formula: $PQ = \sqrt{(6-4)^2 + (8-6)^2} = \sqrt{4 + 4} = 2\sqrt{2}$ units.

Question 22

Given two points $A(2, 3)$ and $B(8, 7)$, find the coordinates of the midpoint of the line segment joining these points.

Accepted Answer

Correct Option: A. Solution: The midpoint formula is given by $$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$. Substituting $A(2, 3)$ and $B(8, 7)$, we get $$\left( \frac{2+8}{2}, \frac{3+7}{2} \right) = (5, 5)$$.

Question 23

Consider three points $P(-3, 4)$, $Q(0, 0)$, and $R(3, -4)$. Which of the following statements is correct?

Accepted Answer

Correct Option: A. Solution: Calculate the distances: $PQ = \sqrt{(-3-0)^2 + (4-0)^2} = 5$, $QR = \sqrt{(0-3)^2 + (0+4)^2} = 5$, and $PR = \sqrt{(-3-3)^2 + (4+4)^2} = 10$. Since $PQ + QR = PR$, the points are collinear.

Question 24

Which of the following statements is true about the midpoint of a line segment?

Accepted Answer

Correct Option: A. Solution: The midpoint divides the line segment into two equal parts, making option a the correct choice.

Question 25

Given three points $A(2, 3)$, $B(5, 7)$, and $C(8, 11)$, determine if these points are collinear.

Accepted Answer

Correct Option: B. Solution: Calculate the distances: $AB = \sqrt{(5-2)^2 + (7-3)^2} = 5$, $BC = \sqrt{(8-5)^2 + (11-7)^2} = 5$, and $AC = \sqrt{(8-2)^2 + (11-3)^2} = 10$. Since $AB + BC = 10$ which equals $AC$, the points are collinear.

Question 26

A point $P(x, y)$ divides the line segment joining $A(-3, 4)$ and $B(5, -2)$ externally in the ratio $2:3$. What are the coordinates of $P$?

Accepted Answer

Correct Option: B. Solution: For external division, $x = \frac{2 	imes 5 - 3 	imes (-3)}{2 - 3} = 1$, $y = \frac{2 	imes (-2) - 3 	imes 4}{2 - 3} = 0$. Thus, $P(1, 0)$.

Question 27

Given points $A(1, 2)$ and $B(5, 6)$, find the coordinates of the midpoint.

Accepted Answer

Correct Option: A. Solution: Using the midpoint formula $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$, the midpoint is $\left(\frac{1 + 5}{2}, \frac{2 + 6}{2}\right) = (3, 4)$.

Question 28

What is the equation of the line on which a point $(x, y)$ is equidistant from the points $(7, 1)$ and $(3, 5)$?

Accepted Answer

Correct Option: B. Solution: The point $(x, y)$ is equidistant from $(7, 1)$ and $(3, 5)$ if $AP = BP$, leading to the equation $x - y = 2$.

Question 29

What is the midpoint of the line segment joining the points $A(1, 7)$ and $B(5, 3)$?

Accepted Answer

Correct Option: A. Solution: The midpoint formula is $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. Thus, the midpoint is $\left(\frac{1 + 5}{2}, \frac{7 + 3}{2}\right) = (3, 5)$.

Question 30

Two points $P(6, 4)$ and $Q(-5, -3)$ are in different quadrants. What is the distance between them?

Accepted Answer

Correct Option: A. Solution: Using the distance formula, $PQ = \sqrt{(6 - (-5))^2 + (4 - (-3))^2} = \sqrt{11^2 + 7^2} = \sqrt{121 + 49} = \sqrt{170}$ units.

Question 31

What is the formula to find the midpoint of a line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$?

Accepted Answer

Correct Option: A. Solution: The midpoint formula is given by $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Question 32

Consider a parallelogram with vertices $A(2, 3)$, $B(8, 3)$, $C(7, -1)$, and $D(1, -1)$. If the diagonals of the parallelogram bisect each other, find the coordinates of the point where the diagonals intersect.

Accepted Answer

Correct Option: C. Solution: The diagonals of a parallelogram bisect each other. The midpoint of diagonal $AC$ is $\left(\frac{2+7}{2}, \frac{3+(-1)}{2}\right) = (4.5, 1)$. The midpoint of diagonal $BD$ is $\left(\frac{8+1}{2}, \frac{3+(-1)}{2}\right) = (4.5, 1)$. Therefore, the coordinates of the point where the diagonals intersect are $(5, 0)$.

Question 33

A point $P(x, y)$ is such that it divides the line segment joining $A(1, -1)$ and $B(4, 3)$ in the ratio $k:1$. If $P$ is at $(3, 1)$, find the value of $k$.

Accepted Answer

Correct Option: A. Solution: Using the section formula: $x = \frac{k 	imes 4 + 1 	imes 1}{k + 1} = 3$, solve for $k$ to get $k = 2$.

Question 34

If the midpoint of a line segment joining points $A(2, 3)$ and $B(x, 7)$ is $(4, 5)$, what is the value of $x$?

Accepted Answer

Correct Option: A. Solution: Using the midpoint formula $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$, we have $\frac{2 + x}{2} = 4$. Solving for $x$, we get $x = 6$.

Question 35

Find the coordinates of the midpoint of the line segment joining $A(6, -4)$ and $B(-2, 8)$.

Accepted Answer

Correct Option: A. Solution: The midpoint formula is $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. Thus, $\left(\frac{6 + (-2)}{2}, \frac{-4 + 8}{2}\right) = (2, 2)$.

Question 36

What is the section formula used to find the coordinates of a point dividing a line segment internally in the ratio $m_1 : m_2$?

Accepted Answer

Correct Option: A. Solution: The section formula for a point dividing a line segment internally in the ratio $m_1 : m_2$ is $\left( \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \right)$.

Question 37

If the midpoint of the line segment connecting points $E(5, -3)$ and $F(x, y)$ is $(2, 1)$, what are the coordinates of $F$?

Accepted Answer

Correct Option: A. Solution: Using the midpoint formula, $$\left( \frac{5+x}{2}, \frac{-3+y}{2} \right) = (2, 1)$$. Solving these equations, $x = -1$ and $y = 5$. Thus, $F$ is $(-1, 5)$.

Question 38

If the point $P(4, y)$ divides the line segment joining $A(2, 3)$ and $B(8, 5)$ in the ratio $1:2$, find the value of $y$.

Accepted Answer

Correct Option: B. Solution: Using the section formula for $y$: $y = \frac{1 	imes 5 + 2 	imes 3}{1 + 2} = 3.67$.

Question 39

In a parallelogram, if the diagonals bisect each other, and one diagonal is between points $C(1, 2)$ and $D(5, 6)$, what is the equation of the line that represents this diagonal?

Accepted Answer

Correct Option: A. Solution: The slope of the line joining points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$. For points $C(1, 2)$ and $D(5, 6)$, the slope $m = \frac{6 - 2}{5 - 1} = 1$. Using the point-slope form $y - y_1 = m(x - x_1)$ with point $C(1, 2)$, we get $y - 2 = 1(x - 1)$, which simplifies to $y = x + 1$.

Question 40

If the diagonals of a parallelogram bisect each other, what are the coordinates of the midpoint of the diagonal joining points $A(2, 3)$ and $B(6, 7)$?

Accepted Answer

Correct Option: A. Solution: The midpoint of a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. Thus, the midpoint of the diagonal joining $A(2, 3)$ and $B(6, 7)$ is $\left(\frac{2 + 6}{2}, \frac{3 + 7}{2}\right) = (4, 5)$.

Question 41

What is the distance between the points $A(4, 0)$ and $B(6, 0)$ on the x-axis?

Accepted Answer

Correct Option: A. Solution: Since both points lie on the x-axis, the distance is simply the difference in their x-coordinates: $6 - 4 = 2$ units.

Question 42

For points $X(1, 2)$, $Y(4, 5)$, and $Z(7, 8)$, determine the nature of these points.

Accepted Answer

Correct Option: A. Solution: Calculate the distances: $XY = \sqrt{(4-1)^2 + (5-2)^2} = 3\sqrt{2}$, $YZ = \sqrt{(7-4)^2 + (8-5)^2} = 3\sqrt{2}$, and $XZ = \sqrt{(7-1)^2 + (8-2)^2} = 6\sqrt{2}$. Since $XY + YZ = XZ$, the points are collinear.

Question 43

Find the ratio in which the y-axis divides the line segment joining the points $(5, -6)$ and $(-1, -4)$.

Accepted Answer

Correct Option: A. Solution: Using the section formula, the x-coordinate of the point of division is 0. Solving $\frac{5k - 1}{k + 1} = 0$ gives $k = 5$. Hence, the ratio is 5:1.

Question 44

If a point $P(3, 4)$ is located on the Cartesian plane, what is its distance from the origin?

Accepted Answer

Correct Option: A. Solution: The distance from the origin is calculated using the formula $\sqrt{x^2 + y^2}$. For $P(3, 4)$, it is $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ units.

Question 45

Find the relation between $x$ and $y$ such that the point $(x, y)$ is equidistant from the points $(2, -1)$ and $(8, 3)$.

Accepted Answer

Correct Option: A. Solution: The perpendicular bisector of the segment joining $(2, -1)$ and $(8, 3)$ is the locus of points equidistant from both. The midpoint is $\left(\frac{2+8}{2}, \frac{-1+3}{2}\right) = (5, 1)$. The slope of the line joining the points is $\frac{3 - (-1)}{8 - 2} = \frac{2}{3}$, thus the perpendicular slope is $-\frac{3}{2}$. Using the point-slope form, $y - 1 = -\frac{3}{2}(x - 5)$, simplifying gives $x - 2y = 4$.

Question 46

Given the points $A(2, -2)$ and $B(-7, 4)$, find the coordinates of the points that trisect the line segment $AB$.

Accepted Answer

Correct Option: A. Solution: The points of trisection divide the segment in the ratios 1:2 and 2:1. Applying the section formula, the coordinates are $(-1, 0)$ and $(-4, 2)$.

Question 47

Given two points $A(2, 3)$ and $B(5, 7)$, what is the distance between them using the distance formula?

Accepted Answer

Correct Option: B. Solution: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Substituting the given points, we have $\sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Question 48

If a point $(x, y)$ is equidistant from $(3, 6)$ and $(-3, 4)$, what is the relation between $x$ and $y$?

Accepted Answer

Correct Option: A. Solution: For the point $(x, y)$ to be equidistant from $(3, 6)$ and $(-3, 4)$, the relation is $x + y = 1$.

Question 49

A point $(x, y)$ is equidistant from points $P(1, 3)$ and $Q(5, 7)$. Which of the following is the correct relation between $x$ and $y$?

Accepted Answer

Correct Option: B. Solution: For a point $(x, y)$ to be equidistant from $P(1, 3)$ and $Q(5, 7)$, the perpendicular bisector of $PQ$ must be satisfied. The midpoint of $PQ$ is $\left(\frac{1+5}{2}, \frac{3+7}{2}\right) = (3, 5)$. The slope of $PQ$ is $1$, so the perpendicular slope is $-1$. The equation of the line is $x - y = 4$.

Question 50

The coordinates of the points dividing a line segment into three equal parts are found using the midpoint formula.

Accepted Answer

Answer: False. Solution: The midpoint formula is used for dividing a line segment into two equal parts. For trisection, the section formula is used with appropriate ratios.

Question 51

The diagonals of a parallelogram bisect each other.

Accepted Answer

Answer: True. Solution: In a parallelogram, each diagonal divides the parallelogram into two congruent triangles, thus bisecting each other.

Question 52

The coordinates of a point $P(x, y)$ that divides a line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m_1:m_2$ are given by $x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}$ and $y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2}$.

Accepted Answer

Answer: True. Solution: The section formula provides the coordinates of point $P(x, y)$ dividing the line segment internally in the ratio $m_1:m_2$ as $x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}$ and $y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2}$.

Question 53

In a parallelogram, the diagonals bisect each other at the midpoint.

Accepted Answer

Answer: True. Solution: The property of a parallelogram is that its diagonals bisect each other, meaning they intersect at their midpoints.

Question 54

The relation $x - y = 2$ represents the condition for a point $(x, y)$ to be equidistant from the points $(7, 1)$ and $(3, 5)$.

Accepted Answer

Answer: True. Solution: The given relation $x - y = 2$ is derived by equating the distances from the point $(x, y)$ to $(7, 1)$ and $(3, 5)$, which are equidistant.

Question 55

Three points are collinear if the sum of the distances between two pairs of points equals the distance between the third pair.

Accepted Answer

Answer: True. Solution: The concept of collinearity involves checking if the sum of the distances between two pairs of points equals the distance between the third pair, confirming they lie on the same straight line.

Question 56

Three points are collinear if the sum of the distances between two pairs of points equals the distance between the third pair.

Accepted Answer

Answer: True. Solution: Collinearity of points can be verified by checking if the sum of the distances between two pairs of points equals the distance between the third pair.

Question 57

The points of trisection of a line segment joining points $A(2, -2)$ and $B(-7, 4)$ are $(-1, 0)$ and $(-4, 2)$.

Accepted Answer

Answer: True. Solution: Using the section formula, the coordinates of the points dividing the line segment in the ratio 1:2 and 2:1 are $(-1, 0)$ and $(-4, 2)$ respectively.

Question 58

The midpoint of a line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is calculated using the formula $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Accepted Answer

Answer: True. Solution: The formula for the midpoint is derived by averaging the x-coordinates and the y-coordinates of the two points.

Question 59

The section formula cannot be used to find the coordinates of a point dividing a line segment externally.

Accepted Answer

Answer: False. Solution: The section formula can be adapted to find the coordinates of a point dividing a line segment externally by considering the ratio as negative.

Question 60

The coordinates of the midpoint of a line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ can be found using the section formula by setting the ratio $m_1:m_2$ to $1:1$.

Accepted Answer

Answer: True. Solution: Setting the ratio $m_1:m_2$ to $1:1$ in the section formula gives the midpoint formula: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Question 61

The midpoint of a line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by the formula $$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$.

Accepted Answer

Answer: True. Solution: The formula for the midpoint of a line segment joining two points is correctly given by $$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$.

Question 62

If a point $P(x, y)$ divides a line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ externally in the ratio $m_1:m_2$, the section formula remains the same as for internal division.

Accepted Answer

Answer: False. Solution: The section formula for external division differs from internal division, and it is not covered in the current class level.

Question 63

If a point $P(x, y)$ divides a line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $1:1$, then $P$ is the midpoint of the segment.

Accepted Answer

Answer: True. Solution: When a point divides a line segment in the ratio $1:1$, it is the midpoint. The coordinates of the midpoint are given by $x = \frac{x_1 + x_2}{2}$ and $y = \frac{y_1 + y_2}{2}$.

Question 64

The section formula can be used to find the coordinates of a point that divides a line segment externally.

Accepted Answer

Answer: False. Solution: The section formula discussed in the chapter is used for finding the coordinates of a point dividing a line segment internally. External division is covered in higher classes.

Coordinate Geometry

Learning Objectives

Learning Objectives

Revision Notes & Summary

Chapter Notes

Distance Formula

Section Formula

Midpoint Formula

Collinearity of Points

Trisection of a Line Segment

Equidistant Points

Parallelogram Diagonal Bisection

Exam Tips & Common Mistakes

Common Mistakes & Exam Tips

Distance Formula

Section Formula

Midpoint Formula

Collinearity of Points

Trisection of a Line Segment

Equidistant Points

Parallelogram Diagonal Bisection

AI Learning Assistant

Practice Test – MCQs, True/False

Experience the StudyTunnel Method

Step 1: Chapter Practice

Step 2: Term Boss Exam

Step 3: Redemption Arena

Practice, Analyze & Improve 🚀

Multiple Choice Questions

Q1.A point P(x,y)P(x, y)P(x,y) divides the line segment joining A(3,6)A(3, 6)A(3,6) and B(−3,4)B(-3, 4)B(−3,4) in the ratio 2:3. What is the value of xxx?

Correct Answer: A

Q2.A point P(x,y)P(x, y)P(x,y) divides the line segment joining A(2,3)A(2, 3)A(2,3) and B(6,7)B(6, 7)B(6,7) in the ratio 3:13:13:1. What are the coordinates of PPP?

Correct Answer: B

Q3.If a point PPP divides the line segment joining A(2,−2)A(2, -2)A(2,−2) and B(−7,4)B(-7, 4)B(−7,4) in the ratio 1:2, what are the coordinates of PPP?

Correct Answer: A

Q4.What is the length of the line segment joining the points (−3,4)(-3, 4)(−3,4) and (3,−4)(3, -4)(3,−4)?

Correct Answer: B

Q5.If a point P(x,y)P(x, y)P(x,y) divides the line segment joining A(−3,4)A(-3, 4)A(−3,4) and B(5,−2)B(5, -2)B(5,−2) externally in the ratio 2:32:32:3, what is the x-coordinate of PPP?

Correct Answer: B

Q6.Find the coordinates of a point on the y-axis which is equidistant from the points (6,5)(6, 5)(6,5) and (−4,3)(-4, 3)(−4,3).

Correct Answer: C

Q7.Given points A(1,5)A(1, 5)A(1,5), B(2,3)B(2, 3)B(2,3), and C(−2,−11)C(-2, -11)C(−2,−11), determine if they are collinear.

Correct Answer: A

Q8.What is the formula to calculate the distance between two points P(x1,y1)P(x_1, y_1)P(x1​,y1​) and Q(x2,y2)Q(x_2, y_2)Q(x2​,y2​) in a Cartesian plane?

Correct Answer: A

Q9.Given the points A(2,3)A(2, 3)A(2,3) and B(10,7)B(10, 7)B(10,7), find the coordinates of the point P(x,y)P(x, y)P(x,y) that divides the line segment ABABAB internally in the ratio 3:23:23:2.

Correct Answer: A

Q10.A point QQQ divides the line segment joining A(2,−2)A(2, -2)A(2,−2) and B(−7,4)B(-7, 4)B(−7,4) in the ratio 2:1. What are the coordinates of QQQ?

Correct Answer: A

Q11.Which of the following points is the midpoint of the line segment joining A(4,−3)A(4, -3)A(4,−3) and B(8,5)B(8, 5)B(8,5)?

Correct Answer: A

Q12.In a coordinate plane, if the distance between the origin (0,0)(0, 0)(0,0) and a point (x,y)(x, y)(x,y) is 101010 units, what could be the possible coordinates of the point?

Correct Answer: A

Q13.Consider two points A(4,2)A(4, 2)A(4,2) and B(8,10)B(8, 10)B(8,10). Find the equation of the line such that any point (x,y)(x, y)(x,y) on this line is equidistant from both AAA and BBB.

Correct Answer: B

Q14.The midpoint of a line segment is (3,4)(3, 4)(3,4). If one endpoint is (1,2)(1, 2)(1,2), what is the other endpoint?

Correct Answer: A

Q15.A line segment is divided by a point PPP such that the ratio of the segments is 1:11:11:1. If the endpoints of the line segment are C(4,−2)C(4, -2)C(4,−2) and D(10,6)D(10, 6)D(10,6), what are the coordinates of point PPP?

Correct Answer: A

Q16.What is the formula to find the coordinates of a point P(x,y)P(x, y)P(x,y) that divides a line segment joining two points A(x1,y1)A(x_1, y_1)A(x1​,y1​) and B(x2,y2)B(x_2, y_2)B(x2​,y2​) internally in the ratio m1:m2m_1:m_2m1​:m2​?

Correct Answer: A

Q17.If a point P(x,y)P(x, y)P(x,y) is equidistant from points A(1,2)A(1, 2)A(1,2) and B(4,6)B(4, 6)B(4,6), what is the equation relating xxx and yyy?

Correct Answer: D

Q18.If a point Q(x,y)Q(x, y)Q(x,y) is equidistant from points A(7,1)A(7, 1)A(7,1) and B(3,5)B(3, 5)B(3,5), what is the relationship between xxx and yyy?

Correct Answer: A

Q19.In a parallelogram, the diagonals bisect each other. Given the vertices A(−3,0)A(-3, 0)A(−3,0), B(3,0)B(3, 0)B(3,0), C(2,5)C(2, 5)C(2,5), and D(−4,5)D(-4, 5)D(−4,5), calculate the coordinates of the intersection point of the diagonals.

Correct Answer: A

Q20.A point P(0,y)P(0, y)P(0,y) on the y-axis is equidistant from the points A(3,4)A(3, 4)A(3,4) and B(3,−4)B(3, -4)B(3,−4). What is the value of yyy?

Correct Answer: A

Q21.What is the distance between the points P(4,6)P(4, 6)P(4,6) and Q(6,8)Q(6, 8)Q(6,8) using the distance formula?

Correct Answer: A

Q22.Given two points A(2,3)A(2, 3)A(2,3) and B(8,7)B(8, 7)B(8,7), find the coordinates of the midpoint of the line segment joining these points.

Correct Answer: A

Q23.Consider three points P(−3,4)P(-3, 4)P(−3,4), Q(0,0)Q(0, 0)Q(0,0), and R(3,−4)R(3, -4)R(3,−4). Which of the following statements is correct?

Correct Answer: A

Q24.Which of the following statements is true about the midpoint of a line segment?

Correct Answer: A

Q25.Given three points A(2,3)A(2, 3)A(2,3), B(5,7)B(5, 7)B(5,7), and C(8,11)C(8, 11)C(8,11), determine if these points are collinear.

Correct Answer: B

Q26.A point P(x,y)P(x, y)P(x,y) divides the line segment joining A(−3,4)A(-3, 4)A(−3,4) and B(5,−2)B(5, -2)B(5,−2) externally in the ratio 2:32:32:3. What are the coordinates of PPP?