# Finding Distance of Two Points Learn how to find the distance of two points on the coordinate plane by using the distance formula.

## Step by step guide to find distance of two points

• Distance of two points A $$(x_{1},y_{1})$$ and B $$(x_{2},y_{2})$$
$$\color{blue}{d=\sqrt{(x_2-x_1)^{2}+(y_{2}-y_{1}){^2}}}$$

### Example 1:

Find the distance of two points $$(1,6) \ and \ (4,2)$$.

Solution:

Use distance of two points formula: $$\color{blue}{d=\sqrt{(x_2-x_1)^{2}+(y_{2}-y_{1}){^2}}}$$
$$(x_{1},y_{1})=(1,6)$$ and $$(x_{2},y_{2})=(4,2)$$. Then: →
$$d=\sqrt{4−(1))^{2}+(2−6)^{2}}=\sqrt{(3)^{2}+(−4)^{2}}=\sqrt{9+16}=\sqrt{25}=5→d=5$$

### Example 2:

Find the distance of two points $$(-1,5) \ and \ (-3,-6)$$.

Solution:

Use distance of two points formula: $$\color{blue}{d=\sqrt{(x_2-x_1)^{2}+(y_{2}-y_{1}){^2}}}$$
$$(x_{1},y_{1})=(-1,5)$$ and $$(x_{2},y_{2})=(-3,-6)$$. Then: $$→$$
$$d=\sqrt{-3−(-1))^{2}+(-6−(5))^{2}}=\sqrt{(-2)^{2}+(−11)^{2}}=\sqrt{4+121}=\sqrt{125}=5\sqrt{5}$$ Then: $$d=5 \sqrt{5}$$

### Example 3:

Find the distance of two points $$(0,8) \ and \ (-4,5)$$.

Solution:

Use distance of two points formula: $$\color{blue}{d=\sqrt{(x_2-x_1)^{2}+(y_{2}-y_{1}){^2}}}$$
$$(x_{1},y_{1})=(0,8)$$ and $$(x_{2},y_{2})=(−4,5)$$. Then: $$d=\sqrt{(x_{1}−x_{2})^2+(y_{1}−y_{2})^2}→$$
$$d=\sqrt{(0−(−4))^2+(8−5)^2}=\sqrt{(4)^2+(3)^2}=\sqrt{16+9}=\sqrt{25}=5→d=5$$

### Example 4:

Find the distance of two points $$(4,2) \ and \ (-5,-10)$$.

Solution:

Use distance of two points formula: $$\color{blue}{d=\sqrt{(x_2-x_1)^{2}+(y_{2}-y_{1}){^2}}}$$
$$(x_{1},y_{1})=(4,2)$$ and $$(x_{2},y_{2})=(−5,-10)$$. Then: $$d=\sqrt{(x_{1}−x_{2})^2+(y_{1}−y_{2})^2}→$$
$$d=\sqrt{(4−(−5))^2+(2−(-10))^2}=\sqrt{(9)^2+(12)^2}=\sqrt{81+144}=\sqrt{225}=25→d=15$$

## Exercises

### Find the distance between each pair of points.

1. $$\color{blue}{(2, –1), ( 1, – 1)}$$
2. $$\color{blue}{ (6, 4), (– 2, 10)}$$
3. $$\color{blue}{(– 8, – 5), (– 5, -1)}$$
4. $$\color{blue}{(– 6, – 10), (– 2, – 10)}$$
5. $$\color{blue}{(4, – 6), (9, 6)}$$
6. $$\color{blue}{(– 6, – 7), (2, 8)}$$

1. $$\color{blue}{1}$$
2. $$\color{blue}{10}$$
3. $$\color{blue}{5}$$
4. $$\color{blue}{4}$$
5. $$\color{blue}{13}$$
6. $$\color{blue}{17}$$ 