Finding the section formula and distance formula in geometry is a very important topic for grade 10 students. Here we are going to learn the importance of the distance formula in coordinate geometry and what is meant by distance formula. The distance of the x-axis is known as the x-coordinate and the distance of the y-axis is known as the y-coordinate. Let us discuss more the distance formula in geometry with related examples.

What is the Formula of Distance in Coordinate Geometry?

The distance formula in geometry helps to find the distance between the two points. In the case of algebraic expression, it is used to calculate the distance between pairs of points required in the coordinates. The distance is equal to the square root of the sum of squares of the coordinates. In coordinate geometry, the distance formula is the calculation of the distance from one point to another point in the plane.

• Distance = √(x2 – x1)² + (y2 – y1)²
• d = |ax1 + by1 +c | /√a² + b²

Distance Formula Examples

Example 1.
Find the distance between the points (3, 4), and (2, 6).
Solution:
The given two points are (3,4) and (2,6)
Using the Euclidean distance formula,
Distance = √(x2 – x1)² + (y2 – y1)²
= √(5 – 3)² + (6 – 4)²
= √(2)² + (2)²
√4 + 4
√8

The distance from the given points to the given points is √8.

Example 2.
Find the distance from the point (2, 5) to the line 2x – 3y = 6
Solution:
The given point is, (x1,x2) = (2,5)
The given line can be written as 2x – 3y – 6 = 0.
Comparing this with ax + by + c = 0, we get a = 2, b = -3, and c = -6.
Using the distance formula to find the distance from a point to a line,
d = |ax1 + by1 +c | /√a² + b²
d = |2(2) + (-3)(5) -6| / √2² + 5²
d = |4 + 15 -6| /4 + 25
d = | 13 | /√29
d = 13 / √29
The distance from the given point to the given line = 13 / √29 units

Example 3.
Calculate the distance between A(6,4) and B on the x- axis whose abscissa is 11.
Solution:
Here B is (11,0)
Distance formula
AB = √(x2 – x1)² + (y2 – y1)²
= √(11 – 6)² + (0-4)²
= √5² + (-4)²
= √25 + 16
= √41
The distance of the given points is √41.

Example 4.
Find the distance between the point (1,3) and the midpoint of the line segment joining (2,4) (4,6).
Solution:
We first find the coordinates of the midpoint M of the segment joining (2, 4) and (4, 6)
M = [ (2 + 4) / 2 , (4 + 6) / 2 ]
= (3, 5)
We now use the distance formula to find the distance between the points (1, 3) and (3, 5)
D = √ [ (3 – (1))² + (5 – (3))²]
= √ (1² + 2²)
= √5
The distance of the two points is √5

Example 5.
Find x and y if (2, 5) is the midpoint of points (x, y) and (-5, 6).
Solution:
We know that
Midpoint formula.
(2, 5) = [ (x + (-5)) / 2 , (y + 6) / 2 ]
Equate the coordinates
2 = (x – 5) / 2 and 5 = (y + 6) / 2
4 = x – 5 and 10 = y + 6
x = 4 + 5 and 10 – 6 = y
x = 9 and y = 4.
Solve for x and y
x = 9 and y = 4

FAQs on Distance Formula in Geometry

1. What does the distance formula prove?

The distance formula reveals that the distance between any two points in a plane is equal to the square root of the sum of squares of differences of the coordinates.

2. What is the distance between points?

The distance between two points is defined as the length of the straight line connecting these points in the coordinate plane.

3. Why is the distance formula important?

The distance formula is a formula used to find the distance between two distinct points on a plane. Finding the distance between two distinct points on a plane is the same as finding the hypotenuse of a right triangle.