Practice Problem on Distance Formula - GeeksforGeeks (2024)

Distance Formula is an important concept in coordinate geometry to find distance between two points or a point and a line or between two lines. This article will explain concepts related to Distance Formula and presents solved and unsolved questions based on them. These questions are essential for students for better clarity and excel in their exam

Important Concepts Related to Distance Formula

Following are some important concepts related to distance formula

Distance between two points (x₁, y₁) and (x₂, y₂) is

• d = √(x₂ – x₁)² + (y₂ – y₁)²

Midpoint Formula:

• Midpoint = ((x₁ + x₂)/2 , (y₁ + y₂)/2)

Distance between a point and a line:

The distance between a point (x₀, y₀) and a line Ax + By + C = 0 is:

• Distance= ∣Ax₀ + By₀ + C∣/√A² + B²

Distance between parallel lines:

If two lines have equations Ax + By + C₁ = 0 and Ax + By + C₂ = 0, then the distance between them is:

• Distance= ∣C₁ − C₂∣/√A² + B²

Practice Questions on Distance Formula with Solution

Example 1. Given two points A(3, 4) and B(7, 9), find the distance between them.

Solution:

To find the distance between two points A(3, 4) and B(7, 9), we use the distance formula:

d = √(7 – 3)² + (9 – 4)²

= √16+25

= √41

So, the distance between two points A(3, 4) and B(7, 9) is √41.

Example 2. Find the midpoint of the line segment joining the points P(2, 5) and Q(8, -3).

Solution:

The midpoint of a line segment PQ with endpoints P(x₁, y₁) and Q(x₂, y₂) is given by the midpoint formula:

Midpoint = (x1 + x2)/2 , (y1 + y2)/2

= (2 + 8)/2, (5 + (-3))/2

= 5, 1

So, the midpoint of the line segment joining the points P(2, 5) and Q(8, -3) is (5, 1).

Example 3. Determine the distance between the point (4, -1) and the line 3x + 4y – 7 = 0.

Solution:

The distance between a point (x₀, y₀) and a line Ax + By + C = 0 is given by:

distance = ∣Ax0 +By0 + C∣/√A² + B²

= ∣3(4) + 4(−1) − 7∣/√3² + 4²

= ∣12−4−7∣ / √9 + 16

= ∣1∣ / √25

= 1/5

So, the distance between the point (4, -1) and the line 3x + 4y – 7 = 0 is 1/5.

Example 4. What is the distance between the point (-1, 6) and the line 2x – 3y + 5 = 0?

Solution:

The distance between a point (x₀, y₀) and a line Ax + By + C = 0 is given by:

See Also

11.2: Distance and Midpoint Formulas and CirclesMidpoint formula | Analytic geometry (article) | Khan Academy11.1 Distance and Midpoint Formulas; Circles - Intermediate Algebra 2e | OpenStaxSection 1.1 The Distance and Midpoint Formulas

distance = ∣Ax0 +By0 + C∣/√A2 + B2

= ∣2(−1) −3(6) + 5∣/√2² + (-3)²

= ∣−2 − 18 + 5∣ / √4 + 9

= ∣−15∣ / √13

= 15/√13

So, the distance between the point (-1, 6) and the line 3x + 4y – 7 = 0 is 15/√13.

Example 5. Find the distance between the parallel lines 2x + 3y – 4 = 0 and 2x + 3y + 6 = 0.

Solution:

To find the distance between the parallel lines 2x + 3y − 4 = 0 and 2x + 3y + 6 = 0, we use the formula:

distance = ∣C2 − C1∣/√A² + B²

Plugging in the values, we get:

distance = ∣6 − (−4)∣/√2² + 3²

​= ∣10∣/√13

= 10/√13

So, the distance between the two parallel lines is 10/√13.

Example 6. Calculate the distance between the parallel lines 4x – 3y – 9 = 0 and 4x – 3y + 7 = 0.

Solution:

To find the distance between the parallel lines 4x – 3y − 9 = 0 and 4x + 3y + 7 = 0, we use the formula:

distance = ∣C2 − C1∣/√A² + B²

Plugging in the values, we get:

distance = ∣7 − (-9)∣/√4²+ 3²

​= ∣16∣/√25

= 16/5

So, the distance between the two parallel lines is 16/5.

Example 7. If A(-2, 1) and B(3, -4) are two points, find the distance between them.

Solution:

To find the distance between two points A(-2, 1) and B(3, -4), we use the distance formula:

d = √(3 – (-2))² + (-4 – 1)²

= √25 + 25

= 5√2

So, the distance between two points A(-2, 1) and B(3, -4) is 5√2.

Example 8. Determine the midpoint of the line segment joining the points C(5, -2) and D(-3, 7).

Solution:

The midpoint of a line segment PQ with endpoints P(x₁, y₁) and Q(x₂, y₂) is given by the midpoint formula:

See Also

Distance formula review | Analytic geometry (article) | Khan Academy

Midpoint = (x1 + x2)/2 , (y1 + y2)/2

= (5 + (-3))/2, (-2 + 7)/2

= 2, 5

So, the midpoint of the line segment joining the points P(5, -2) and Q(-3, 7) is (2, 5).

Example 9. What is the distance between the point (1, 3) and the line 5x – 2y + 8 = 0?

Solution:

The distance between a point (x₀, y₀) and a line Ax + By + C = 0 is given by:

distance = ∣Ax0 +By0 + C∣/√A2 + B2

= ∣5(1) − 2(3) + 8∣/√5² + (-2)²

= ∣5 − 6 + 8∣ / √25 + 4

= ∣7∣ / √29

= 7/√29

So, the distance between the point (4, -1) and the line 3x + 4y – 7 = 0 is 7/√29.

Example 10. Find the distance between the parallel lines 3x – 4y + 6 = 0 and 3x – 4y – 2 = 0.

Solution:

To find the distance between the parallel lines 3x – 4y + 6 = 0 and 3x – 4y – 2 = 0, we use the formula:

distance = ∣C2 − C1∣/√A² + B²

Plugging in the values, we get:

distance = ∣-2 − (6)∣/√3² + 4²

​= ∣-8∣/√25

= 8/5

So, the distance between the two parallel lines is 8/5.

Practice Problem on Distance Formula

Q1. Find the distance between the points (3, 4) and (-1, 2).

Q2. Determine the midpoint of the line segment with endpoints (5, -3) and (-7, 8).

Q3. Calculate the distance between the point (2, -1) and the line 3x + 4y – 5 = 0.

Q4. Find the distance between the parallel lines 2x + 3y – 7 = 0 and 2x + 3y + 9 = 0.

Q5. Determine the distance between the points (-2, 5) and (1, -3).

Q6. Calculate the midpoint of the line segment with endpoints (-4, 6) and (8, -2).

Q7. Find the distance between the point (3, 7) and the line 4x – 2y + 10 = 0.

Q8. Determine the distance between the parallel lines 3x + 2y – 6 = 0 and 3x + 2y + 12 = 0.

Q9. Calculate the distance between the points (0, -1) and (5, 4).

Q10. Find the midpoint of the line segment with endpoints (-3, 2) and (7, -6).

FAQs on Practice Problem on Distance Formula

Does distance have a negative value?

No, distance does not have a negative value. It’s value is always positive or zero.

Why do we need to use Distance Formula?

We need to use Distance Formula to measure the distance between two points.

Can we calculate speed with the help of Distance?

Yes, we can calculate speed with the help of Distance by applying the following formula, Speed = Distance/Time

What is the SI unit used for distance?

The SI unit used for distance is metre (m).