Class 10 Maths Chapter 7 - Coordinate Geometry | INSTALEARN

Question 1

Assertion (A): The distance between two points and in the coordinate plane is given by the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
Reason (R): The distance between two points on the coordinate plane is the length of the line segment connecting them.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The distance formula is derived from the Pythagorean Theorem, where the distance between two points is the length of the hypotenuse of a right triangle.

Question 2

Assertion (A): The coordinates of the midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ are $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ .
Reason (R): The midpoint of a line segment divides it into two equal parts.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula for the midpoint is derived by averaging the x-coordinates and the y-coordinates of the endpoints, which splits the line segment into two equal halves.

Question 3

Assertion (A): The slope of a vertical line is always zero.
Reason (R): The slope of a line measures the rate of change of $y$ with respect to $x$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: The slope of a vertical line is undefined because there is no change in $x$ , which would require division by zero.

Question 4

Assertion (A): The coordinates of the centroid of a triangle with vertices $(x_1, y_1)$ , $(x_2, y_2)$ , and $(x_3, y_3)$ are $\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$ .
Reason (R): The centroid of a triangle divides each median in a 2:1 ratio.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The centroid formula is derived from the averages of the coordinates of the vertices, and the centroid divides each median in a 2:1 ratio.

Question 5

Assertion (A): The slope of a line passing through the points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\frac{y_2 - y_1}{x_2 - x_1}$ .
Reason (R): The slope of a line is the ratio of the vertical change to the horizontal change between two points.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The slope is the change in the y-coordinates divided by the change in the x-coordinates, which is the definition of the slope.

Question 6

Assertion (A): If the distance between two points is zero, then the points must coincide.
Reason (R): The distance between two distinct points is always positive.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: If the distance is zero, the two points must be the same, because any distinct points would have a positive distance.

Question 7

Assertion (A): The area of a triangle with vertices $(x_1, y_1)$ , $(x_2, y_2)$ , and $(x_3, y_3)$ is given by $\frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$ .
Reason (R): The area of a triangle is half the absolute value of the determinant formed by the coordinates of its vertices.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula for the area of a triangle using its vertices comes from the determinant method and provides the area in terms of the coordinates of the vertices.

Question 8

Assertion (A): If the slope of a line is zero, the line is horizontal.
Reason (R): A horizontal line has no vertical change between any two points.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: A line with zero slope means there is no vertical change, so the line is horizontal.

Question 9

Assertion (A): The equation of a line in slope-intercept form is written as $y = mx + c$ .
Reason (R): In this equation, $m$ represents the slope and $c$ represents the y-intercept.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The slope-intercept form explicitly shows the slope and y-intercept, which determine the properties of the line.

Question 10

Assertion (A): The slope of the x-axis is zero.
Reason (R): The x-axis is a horizontal line, and all horizontal lines have zero slope.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The x-axis has no vertical change, so the slope is zero, which applies to all horizontal lines.

Question 11

Assertion (A): The coordinates of a point can be negative.
Reason (R): The coordinate plane is divided into four quadrants, which allow for both positive and negative values.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The coordinate plane comprises four quadrants where the x and y coordinates can be either positive or negative.

Question 12

Assertion (A): The formula for the section formula is used to find the coordinates of a point that divides a line segment in a given ratio.
Reason (R): The section formula is given by $\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The section formula provides the coordinates of a point dividing the line segment in a given ratio, confirming the assertion.

Question 13

Assertion (A): If two lines are parallel, they have the same slope.
Reason (R): Lines with different slopes will eventually intersect at some point.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Parallel lines share the same slope and will not intersect, confirming the relationship between slope and line behavior.

Question 14

Assertion (A): The area of a rectangle can be calculated using the coordinates of its vertices.
Reason (R): The formula for the area is given by $|x_2 - x_1| \times |y_2 - y_1|$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The area formula calculates the product of the length and width, which can be derived from the coordinates of the rectangle's vertices.

Question 15

Assertion (A): A line parallel to the y-axis has an equation of the form $x = a$ .
Reason (R): The line does not change in the y-direction, hence it has no slope.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: A vertical line represented by $x = a$ indicates that it has a constant x-value and extends indefinitely in the y-direction, hence it has no slope.

Question 16

Assertion (A): The equation of a circle in the Cartesian plane is $(x - h)^2 + (y - k)^2 = r^2$ .
Reason (R): In this equation, $(h, k)$ is the center, and $r$ is the radius.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The standard form of the circle’s equation clearly defines the center and radius, confirming both statements.

Question 17

Assertion (A): The sum of the slopes of two perpendicular lines is $-1$ .
Reason (R): The product of the slopes of two perpendicular lines is always equal to $-1$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: The product of the slopes being $-1$ indicates that the lines are perpendicular, not the sum.

Question 18

Assertion (A): The coordinates of the point of intersection of two lines can be found by solving their equations simultaneously.
Reason (R): This method is known as the graphical method of solving linear equations.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: C) A is true, but R is false.
Explanation: The point of intersection can be found by solving simultaneously, but this is typically called the algebraic method, not the graphical method.

Question 19

Assertion (A): A triangle is determined uniquely by its three sides.
Reason (R): This property is known as the Side-Side-Side (SSS) criterion for congruence.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The SSS criterion states that if the lengths of the sides of a triangle are known, the triangle is uniquely determined.

Question 20

Assertion (A): If a line intersects the x-axis at point A and the y-axis at point B, the coordinates of point A are of the form $(x, 0)$ .
Reason (R): The x-coordinate represents the position along the x-axis while the y-coordinate is zero for any point on the x-axis.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: By definition, any point on the x-axis has a y-coordinate of zero.

Question 21

Assertion (A): The distance between two points can never be negative.
Reason (R): Distance is defined as the absolute value of the difference between two points.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The definition of distance ensures it is always a non-negative value, hence never negative.

Question 22

Assertion (A): The coordinates of a point on the line $y = 2x + 3$ can be found by substituting a value for $x$ .
Reason (R): This equation represents a linear relationship between $x$ and $y$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Substituting a value for $x$ gives a corresponding $y$ value, confirming the linear relationship.

Question 23

Assertion (A): The midpoint of a line segment can be calculated using the coordinates of its endpoints.
Reason (R): The midpoint formula is given by $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The midpoint formula accurately describes how to find the midpoint using the coordinates of the endpoints.

Question 24

Assertion (A): The slope of a horizontal line is zero.
Reason (R): A horizontal line does not rise or fall as it extends along the x-axis.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The absence of change in y-values for a horizontal line results in a slope of zero.

Question 25

Assertion (A): The equation of a line in slope-intercept form is given by $y = mx + c$ .
Reason (R): Here, $m$ represents the slope of the line, and $c$ represents the y-intercept.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The slope-intercept form clearly defines the components of the line's equation, affirming both statements.

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Class 10 Maths Chapter 7 - Coordinate Geometry