# Distance Formula

The distance between two points is calculated using the formula
$$\sqrt{{(x_{2}-x_{1})^{2}} +(y_{2}-y_{1})^{2}}$$

Example1: Find the distance between the points (1, -3) and the point (5, -8)

Solution: The distance between the points A and B is given by the formula
$$\sqrt{{(x_{2}-x_{1})^{2}} +(y_{2}-y_{1})^{2}}$$
Here A= (1, -3) = (x1, y1) and B= (5, -8) = (x2, y2). Now apply this formula
$$\sqrt{{(5- 1)^{2}} +(-8 + 3)^{2}}$$
$$\sqrt{{(4)^{2}} +(-5)^{2}}$$
$$\sqrt{16 + 25}$$
$$\sqrt{41}$$

Example 2: Find the distance between the points (2, -7) and (3, 2)

Solution: The distance between the points A and B is given by the formula
$$\sqrt{{(x_{2}-x_{1})^{2}} +(y_{2}-y_{1})^{2}}$$
Here A= (1, -3) = (x1, y1) and B= (5, -8) = (x2, y2). Now apply this formula
$$\sqrt{{(3- 2)^{2}} +(2 + 7)^{2}}$$
$$\sqrt{{(1)^{2}} +(9)^{2}}$$
$$\sqrt{1+ 81}$$
$$\sqrt{82}$$

Example3: Find the distance between the points (a, b) and (-a, -b).

Solution: The distance between the points A and B is given by the formula
$$\sqrt{{(x_{2}-x_{1})^{2}} +(y_{2}-y_{1})^{2}}$$
Here A= (1, -3) = (x1, y1) and B= (5, -8) = (x2, y2). Now apply this formula
$$\sqrt{{(-a- a)^{2}} +(-b -b)^{2}}$$
$$\sqrt{{(-2a)^{2}} +(-2b)^{2}}$$
$$\sqrt{4a^2 +4b^2}$$
$$2 \sqrt{a^2+b^2}$$

Median : A median is a line segment joining a vertex to a midpoint to the opposite side of the triangle.

Centroid : The intersection of 3 medians of the triangle is called the centroid of the triangle

## Centroid Formula

The formula for centroid is given by
$$[ \frac{(x_{1}+x_{2}+x_{3})}{3}, \frac{(y_{1}+y_{2}+y_{3})}{3} ]$$

Examples:

Find the centroid of the triangle with the points (1, 2), (3, 5), (7, 8)

We have the formula
$$[ \frac{(1+3+7)}{3}, \frac{(2+5+8)}{3} ]$$
$$(\frac{11}{3}, \frac{15}{3}) = ((\frac{11}{3}, 5)$$

Find the centroid of the triangle with the points (5, 2), (6, 5), (-7, 8)
We have the formula
$$[ \frac{(5+6-7)}{3}, \frac{(2+5+8)}{3} ]$$
$$(\frac{4}{3}, \frac{15}{3}) = ((\frac{4}{3}, 5)$$

Find the centroid of the triangle with the points (0, 2), (6, 1), (-7, 3)
We have the formula
$$[ \frac{(0+6-7)}{3}, \frac{(2+1+3)}{3} ]$$
$$(\frac{-1}{3}, \frac{7}{3})$$

The above three examples show us how to calculate the centroid.

Ortho Centre: The position where the 3 altitudes of a triangle meet is called the Ortho Centre of the triangle.

The below diagram shows all the above points:

Next Chapters

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