Distance Formula MCQs Quiz | Class 10

Understanding the Distance Formula

The distance formula is a fundamental concept in coordinate geometry used to find the distance between two points in a two-dimensional Cartesian coordinate system. It allows us to measure the length of a line segment connecting any two given points.

The Distance Formula

Given two points A(x1, y1) and B(x2, y2), the distance ‘d’ between them is given by:

d = sqrt((x2 – x1)^2 + (y2 – y1)^2)

Where ‘sqrt’ denotes the square root.

Derivation Idea (Using Pythagorean Theorem)

The distance formula is derived directly from the Pythagorean theorem. Consider two points A(x1, y1) and B(x2, y2).

1. Draw a horizontal line through A and a vertical line through B. These lines intersect at a point C.
2. The coordinates of C will be (x2, y1).
3. Now, we have a right-angled triangle ABC, with the right angle at C.
4. The length of the horizontal side AC is the absolute difference in the x-coordinates: AC = |x2 – x1|.
5. The length of the vertical side BC is the absolute difference in the y-coordinates: BC = |y2 – y1|.
6. According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (AB) is equal to the sum of the squares of the other two sides (AC and BC):

AB^2 = AC^2 + BC^2

AB^2 = (x2 – x1)^2 + (y2 – y1)^2

Taking the square root of both sides gives us the distance formula:

AB = sqrt((x2 – x1)^2 + (y2 – y1)^2)

Using the Distance Formula (Numerical Applications)

To find the distance between two points, follow these steps:

1. Identify the coordinates of the two points as (x1, y1) and (x2, y2).
2. Substitute the values into the formula: d = sqrt((x2 – x1)^2 + (y2 – y1)^2).
3. Perform the subtractions (x2 – x1) and (y2 – y1).
4. Square the results of the subtractions.
5. Add the squared values.
6. Take the square root of the sum.

Example: Find the distance between P(1, 2) and Q(4, 6).

• Let (x1, y1) = (1, 2) and (x2, y2) = (4, 6).
• d = sqrt((4 – 1)^2 + (6 – 2)^2)
• d = sqrt((3)^2 + (4)^2)
• d = sqrt(9 + 16)
• d = sqrt(25)
• d = 5 units.

Key Concepts and Applications

• Distance from Origin: The distance of a point (x, y) from the origin (0, 0) is `sqrt(x^2 + y^2)`.
• Collinearity: Three points A, B, C are collinear if the sum of the distances between two pairs of points equals the distance of the third pair (e.g., AB + BC = AC).
• Types of Triangles: The formula can be used to classify triangles (equilateral, isosceles, scalene) by comparing side lengths, or to check for right-angled triangles using the converse of the Pythagorean theorem.
• Types of Quadrilaterals: It helps to determine if a quadrilateral is a square, rectangle, rhombus, or parallelogram by checking side lengths and diagonals.
• Equidistant Points: Finding a point that is equally distant from two or more given points.

Distance Scenarios Summary

Quick Revision

• The distance formula helps find the length between two points.
• It is derived from the Pythagorean theorem.
• Formula: `d = sqrt((x2 – x1)^2 + (y2 – y1)^2)`.
• Used to identify geometric shapes and check for collinearity.
• Always results in a non-negative value.

Practice Questions

1. Find the distance between the points A(-3, 4) and B(9, -1).
2. Determine if the points (1, 5), (2, 3) and (-2, -11) are collinear.
3. Show that the points (1, 7), (4, 2), (-1, -1) and (-4, 4) are the vertices of a square.
4. Find the point on the x-axis which is equidistant from (2, -5) and (-2, 9).
5. What is the radius of a circle if its center is at (3, 2) and it passes through the point (-5, 6)?