All Exams > Class 9 > Mathematics Class 9 ICSE > All Questions
All questions of Distance Formula for Class 9 Exam
How do you determine the coordinates of the midpoint of a line segment between points \( A(x_1, y_1) \) and \( B(x_2, y_2) \)?- a)\( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
- b)\( (x_1 + x_2, y_1 + y_2) \)
- c)\( \left( \frac{x_1 - x_2}{2}, \frac{y_1 - y_2}{2} \right) \)
- d)\( \left( x_1 - x_2, y_1 - y_2 \right) \)
Correct answer is option 'A'. Can you explain this answer?
How do you determine the coordinates of the midpoint of a line segment between points \( A(x_1, y_1) \) and \( B(x_2, y_2) \)?
a)
\( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
b)
\( (x_1 + x_2, y_1 + y_2) \)
c)
\( \left( \frac{x_1 - x_2}{2}, \frac{y_1 - y_2}{2} \right) \)
d)
\( \left( x_1 - x_2, y_1 - y_2 \right) \)
 | Imk Pathshala answered |
The coordinates of the midpoint of a line segment between two points are found using the formula \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). This formula is essential for various applications, including finding central points in geometric figures.
If two points have coordinates (x, y) and (x, y+5), what is the distance between them?- a)0
- b)5
- c)10
- d)25
Correct answer is option 'B'. Can you explain this answer?
If two points have coordinates (x, y) and (x, y+5), what is the distance between them?
a)
0
b)
5
c)
10
d)
25
 | Sahil Singh answered |
Understanding the Points
To find the distance between the two points (x, y) and (x, y + 5), we need to analyze their coordinates.
- The first point is (x, y).
- The second point is (x, y + 5).
Identifying the Distance Formula
The distance between two points (x1, y1) and (x2, y2) in a Cartesian plane can be calculated using the distance formula:
- Distance = √((x2 - x1)² + (y2 - y1)²)
Applying the Formula
In our case, substituting the coordinates into the formula:
- x1 = x, y1 = y
- x2 = x, y2 = y + 5
Now, plugging these values into the distance formula:
- Distance = √((x - x)² + ((y + 5) - y)²)
This simplifies to:
- Distance = √(0² + (5)²)
Calculating the Distance
- Distance = √(0 + 25)
- Distance = √25
- Distance = 5
Conclusion
Therefore, the distance between the points (x, y) and (x, y + 5) is 5 units.
The correct answer is option 'B'.
To find the distance between the two points (x, y) and (x, y + 5), we need to analyze their coordinates.
- The first point is (x, y).
- The second point is (x, y + 5).
Identifying the Distance Formula
The distance between two points (x1, y1) and (x2, y2) in a Cartesian plane can be calculated using the distance formula:
- Distance = √((x2 - x1)² + (y2 - y1)²)
Applying the Formula
In our case, substituting the coordinates into the formula:
- x1 = x, y1 = y
- x2 = x, y2 = y + 5
Now, plugging these values into the distance formula:
- Distance = √((x - x)² + ((y + 5) - y)²)
This simplifies to:
- Distance = √(0² + (5)²)
Calculating the Distance
- Distance = √(0 + 25)
- Distance = √25
- Distance = 5
Conclusion
Therefore, the distance between the points (x, y) and (x, y + 5) is 5 units.
The correct answer is option 'B'.
What is the relationship between the sides of an isosceles triangle?- a)All sides are equal
- b)Two sides are equal, and the third is different
- c)The sum of any two sides equals the third
- d)No sides are equal
Correct answer is option 'B'. Can you explain this answer?
What is the relationship between the sides of an isosceles triangle?
a)
All sides are equal
b)
Two sides are equal, and the third is different
c)
The sum of any two sides equals the third
d)
No sides are equal
 | Gauri Nambiar answered |
Understanding Isosceles Triangles
An isosceles triangle is a special type of triangle characterized by the relationship between its sides. Let's delve into the unique properties that define an isosceles triangle.
Key Properties of Isosceles Triangles
- Two Equal Sides: In an isosceles triangle, two sides are equal in length. These are referred to as the "legs" of the triangle.
- Different Third Side: The third side, known as the "base," is different in length from the two equal sides.
Why Option 'B' is Correct
- Definition Confirmation: The correct answer is option 'B' because it accurately describes the defining feature of an isosceles triangle: two sides are of equal length while the third side is different.
- Visual Representation: If you visualize an isosceles triangle, you will see that the two equal sides form the apex, and the different side acts as the base.
Other Options Explained
- Option 'A': This refers to an equilateral triangle, where all sides are equal, which is not the case for isosceles triangles.
- Option 'C': This states the triangle inequality theorem but does not specifically relate to the isosceles triangle’s properties.
- Option 'D': This describes a scalene triangle, where no sides are equal, which contradicts the definition of an isosceles triangle.
In summary, the defining characteristic of an isosceles triangle is that it has two equal sides and one different side, making option 'B' the correct choice.
An isosceles triangle is a special type of triangle characterized by the relationship between its sides. Let's delve into the unique properties that define an isosceles triangle.
Key Properties of Isosceles Triangles
- Two Equal Sides: In an isosceles triangle, two sides are equal in length. These are referred to as the "legs" of the triangle.
- Different Third Side: The third side, known as the "base," is different in length from the two equal sides.
Why Option 'B' is Correct
- Definition Confirmation: The correct answer is option 'B' because it accurately describes the defining feature of an isosceles triangle: two sides are of equal length while the third side is different.
- Visual Representation: If you visualize an isosceles triangle, you will see that the two equal sides form the apex, and the different side acts as the base.
Other Options Explained
- Option 'A': This refers to an equilateral triangle, where all sides are equal, which is not the case for isosceles triangles.
- Option 'C': This states the triangle inequality theorem but does not specifically relate to the isosceles triangle’s properties.
- Option 'D': This describes a scalene triangle, where no sides are equal, which contradicts the definition of an isosceles triangle.
In summary, the defining characteristic of an isosceles triangle is that it has two equal sides and one different side, making option 'B' the correct choice.
In finding the circumcentre of a triangle, which step involves using the Distance Formula?- a)Setting distances from circumcentre to vertices equal
- b)Finding the area of the triangle
- c)Determining the lengths of the sides
- d)Calculating the angles of the triangle
Correct answer is option 'A'. Can you explain this answer?
In finding the circumcentre of a triangle, which step involves using the Distance Formula?
a)
Setting distances from circumcentre to vertices equal
b)
Finding the area of the triangle
c)
Determining the lengths of the sides
d)
Calculating the angles of the triangle
| | Imk Pathshala answered |
The Distance Formula is used in the process of setting the distances from the circumcentre to each of the triangle's vertices equal to each other, as this defines the circumcentre's location relative to the triangle's vertices.
How can you verify if points A, B, and C are equidistant from a point P (circumcentre)?- a)Check if PA = PB = PC
- b)Check if PA + PB = PC
- c)Check if AB + AC = BC
- d)Check if the angles at P are equal
Correct answer is option 'A'. Can you explain this answer?
How can you verify if points A, B, and C are equidistant from a point P (circumcentre)?
a)
Check if PA = PB = PC
b)
Check if PA + PB = PC
c)
Check if AB + AC = BC
d)
Check if the angles at P are equal
 | Let's Tute answered |
To verify that points A, B, and C are equidistant from point P (the circumcentre), you check if \( PA = PB = PC \). This equality confirms that P is the circumcentre, as it is defined by being equidistant to all vertices of the triangle.
Which property characterizes a rectangle in terms of its sides?- a)All sides are equal
- b)Only one pair of opposite sides is equal
- c)Opposite sides are equal and diagonals are equal
- d)All angles are acute
Correct answer is option 'C'. Can you explain this answer?
Which property characterizes a rectangle in terms of its sides?
a)
All sides are equal
b)
Only one pair of opposite sides is equal
c)
Opposite sides are equal and diagonals are equal
d)
All angles are acute
| | Let's Tute answered |
A rectangle is characterized by having opposite sides that are equal in length, along with equal diagonals. This property is fundamental in defining rectangles and distinguishing them from other quadrilaterals.
If the coordinates of a point on the x-axis are (5, 0), what can be inferred about its location?- a)It lies below the x-axis
- b)It lies on the y-axis
- c)It lies on the x-axis
- d)It lies above the x-axis
Correct answer is option 'C'. Can you explain this answer?
If the coordinates of a point on the x-axis are (5, 0), what can be inferred about its location?
a)
It lies below the x-axis
b)
It lies on the y-axis
c)
It lies on the x-axis
d)
It lies above the x-axis
| | Let's Tute answered |
The point (5, 0) lies on the x-axis because its y-coordinate is 0. This basic understanding of coordinate positions is essential for graphing and analyzing points in a Cartesian plane.
If the coordinates of point A are (4, 5) and point B are (4, 1), what is the distance between them?- a)10
- b)5
- c)0
- d)4
Correct answer is option 'D'. Can you explain this answer?
If the coordinates of point A are (4, 5) and point B are (4, 1), what is the distance between them?
a)
10
b)
5
c)
0
d)
4
| | Let's Tute answered |
The distance between points A and B is calculated as: \[ \text{Distance} = \sqrt{(4 - 4)^2 + (1 - 5)^2} = \sqrt{0 + 16} = \sqrt{16} = 4. \] This example shows that when two points have the same x-coordinate, the distance is simply the absolute difference of their y-coordinates.
What is the circumradius of a triangle?- a)The distance from the circumcentre to one of the vertices
- b)The length of the altitude
- c)The radius of the incircle
- d)The distance from the centroid to one of the sides
Correct answer is option 'A'. Can you explain this answer?
What is the circumradius of a triangle?
a)
The distance from the circumcentre to one of the vertices
b)
The length of the altitude
c)
The radius of the incircle
d)
The distance from the centroid to one of the sides
 | EduRev Class 9 answered |
The circumradius is defined as the distance from the circumcentre to any of the triangle's vertices. It is a crucial feature in triangle geometry, particularly when discussing circles that circumscribe triangles.
Given the points \( A(1, 2) \) and \( B(3, 8) \), what is the calculated distance between them?- a)8
- b)\( 2\sqrt{10} \)
- c)6
- d)\( 2\sqrt{8} \)
Correct answer is option 'B'. Can you explain this answer?
Given the points \( A(1, 2) \) and \( B(3, 8) \), what is the calculated distance between them?
a)
8
b)
\( 2\sqrt{10} \)
c)
6
d)
\( 2\sqrt{8} \)
| | Let's Tute answered |
The distance is calculated as follows: \[ \text{Distance} = \sqrt{(3 - 1)^2 + (8 - 2)^2} = \sqrt{(2)^2 + (6)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10}. \] This demonstrates the application of the Distance Formula to find the distance between two non-vertical points.
What is the circumcentre of a triangle?- a)The point where the medians intersect
- b)The point where the altitudes meet
- c)The center of the triangle's inscribed circle
- d)The point equidistant from all three vertices
Correct answer is option 'D'. Can you explain this answer?
What is the circumcentre of a triangle?
a)
The point where the medians intersect
b)
The point where the altitudes meet
c)
The center of the triangle's inscribed circle
d)
The point equidistant from all three vertices
| | EduRev Class 9 answered |
The circumcentre of a triangle is defined as the point that is equidistant from all three vertices. This point serves as the center of the circumcircle, which is the circle that passes through all vertices of the triangle.
Which of the following is the correct formula for calculating the distance between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \)?- a)\( (x_2 - x_1) + (y_2 - y_1) \)
- b)\( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
- c)\( \sqrt{(x_2 - x_1)^2 + (y_2 + y_1)^2} \)
- d)\( |x_2 - x_1| + |y_2 - y_1| \)
Correct answer is option 'B'. Can you explain this answer?
Which of the following is the correct formula for calculating the distance between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \)?
a)
\( (x_2 - x_1) + (y_2 - y_1) \)
b)
\( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
c)
\( \sqrt{(x_2 - x_1)^2 + (y_2 + y_1)^2} \)
d)
\( |x_2 - x_1| + |y_2 - y_1| \)
| | Let's Tute answered |
The correct formula for calculating the distance between two points is \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). This formula derives from the Pythagorean theorem, allowing us to find the length of the hypotenuse of a right triangle formed by these points.
What does the distance from the origin (0,0) to a point (x,y) represent?- a)The slope of the line connecting the origin to the point
- b)The radius of the circumcircle
- c)The area of a triangle formed with the origin
- d)The straight line distance from the origin to the point
Correct answer is option 'D'. Can you explain this answer?
What does the distance from the origin (0,0) to a point (x,y) represent?
a)
The slope of the line connecting the origin to the point
b)
The radius of the circumcircle
c)
The area of a triangle formed with the origin
d)
The straight line distance from the origin to the point
| | Let's Tute answered |
The distance from the origin (0,0) to a point (x,y) represents the straight-line distance between these two points, calculated using the formula \( \sqrt{x^2 + y^2} \). This concept is crucial in many applications of geometry.
Which of the following statements is true about points on the x-axis?- a)They have x-coordinates of 0
- b)Their y-coordinates are always zero
- c)They can have any real number as a y-coordinate
- d)Their y-coordinates are always positive
Correct answer is option 'B'. Can you explain this answer?
Which of the following statements is true about points on the x-axis?
a)
They have x-coordinates of 0
b)
Their y-coordinates are always zero
c)
They can have any real number as a y-coordinate
d)
Their y-coordinates are always positive
| | Imk Pathshala answered |
Points on the x-axis have a y-coordinate of 0, which means they are represented as \( (x, 0) \) for any real number x. This characteristic is fundamental in coordinate geometry.
Which equation represents the condition for collinearity of three points A, B, and C?- a)\( AB + AC = AB \)
- b)\( AB + BC = AC \)
- c)\( AB + AC = BC \)
- d)\( AB + BC = AC \)
Correct answer is option 'C'. Can you explain this answer?
Which equation represents the condition for collinearity of three points A, B, and C?
a)
\( AB + AC = AB \)
b)
\( AB + BC = AC \)
c)
\( AB + AC = BC \)
d)
\( AB + BC = AC \)
| | Imk Pathshala answered |
The points A, B, and C are collinear if the sum of the distances \( AB + AC = BC \). This condition means that when the points are plotted, they lie on a straight line, which can be verified using the Distance Formula.
When using the Distance Formula, which of the following is crucial to ensure accuracy?- a)Always calculating in two dimensions
- b)Using integer coordinates only
- c)Rounding coordinates to the nearest whole number
- d)Ensuring correct signs for coordinates
Correct answer is option 'D'. Can you explain this answer?
When using the Distance Formula, which of the following is crucial to ensure accuracy?
a)
Always calculating in two dimensions
b)
Using integer coordinates only
c)
Rounding coordinates to the nearest whole number
d)
Ensuring correct signs for coordinates
| | EduRev Class 9 answered |
When applying the Distance Formula, it is crucial to ensure that the correct signs for coordinates are used, as the position of the points in different quadrants affects the outcome of the distance calculation. This attention to detail is vital for accuracy in geometric applications.
What is the primary purpose of the Distance Formula in coordinate geometry?- a)To measure the distance between two points
- b)To determine the angles in a triangle
- c)To calculate the slope of a line
- d)To find the midpoint of a line segment
Correct answer is option 'A'. Can you explain this answer?
What is the primary purpose of the Distance Formula in coordinate geometry?
a)
To measure the distance between two points
b)
To determine the angles in a triangle
c)
To calculate the slope of a line
d)
To find the midpoint of a line segment
| | Imk Pathshala answered |
The Distance Formula is used to calculate the straight-line distance between two points on a coordinate plane. It provides a method to quantify how far apart two points are, which is essential in various applications of geometry and real-world problems.
For points on the y-axis, what is a defining characteristic of their coordinates?- a)Their x-coordinates are always positive
- b)Their x-coordinates are always zero
- c)They can have any real number as a y-coordinate
- d)They have a y-coordinate of 0
Correct answer is option 'C'. Can you explain this answer?
For points on the y-axis, what is a defining characteristic of their coordinates?
a)
Their x-coordinates are always positive
b)
Their x-coordinates are always zero
c)
They can have any real number as a y-coordinate
d)
They have a y-coordinate of 0
| | Imk Pathshala answered |
Points on the y-axis have an x-coordinate of 0, which means they can be represented as \( (0, y) \) for any real number y. This understanding is important for navigating the Cartesian plane.
If the coordinates of points A, B, and C are (1, 1), (2, 2), and (3, 3), respectively, what can be concluded about these points?- a)They form an equilateral triangle
- b)They form a square
- c)They form a right angle
- d)They are collinear
Correct answer is option 'D'. Can you explain this answer?
If the coordinates of points A, B, and C are (1, 1), (2, 2), and (3, 3), respectively, what can be concluded about these points?
a)
They form an equilateral triangle
b)
They form a square
c)
They form a right angle
d)
They are collinear
| | Imk Pathshala answered |
The points A, B, and C are collinear because they lie on the same straight line, which can be verified by checking that the slope between any two pairs of points is the same. This is a key concept in coordinate geometry.
If point A is at (3, 4) and point B is at (0, 0), what is the distance between them?- a)4
- b)3
- c)5
- d)7
Correct answer is option 'C'. Can you explain this answer?
If point A is at (3, 4) and point B is at (0, 0), what is the distance between them?
a)
4
b)
3
c)
5
d)
7
| | Imk Pathshala answered |
The distance is calculated as follows: \[ \text{Distance} = \sqrt{(0 - 3)^2 + (0 - 4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \] This illustrates the application of the Distance Formula in determining the straight-line distance.
Chapter doubts & questions for Distance Formula - Mathematics Class 9 ICSE 2026 is part of Class 9 exam preparation. The chapters have been prepared according to the Class 9 exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Class 9 2026 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.
Chapter doubts & questions of Distance Formula - Mathematics Class 9 ICSE in English & Hindi are available as part of Class 9 exam. Download more important topics, notes, lectures and mock test series for Class 9 Exam by signing up for free.
Mathematics Class 9 ICSE64 videos|165 docs|28 tests |
