Question

distance formula
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
example: find the distance between $(2, -3)$ and $(7, 2)$
$d = \sqrt{(7 - 2)^2 + (2 - 3)^2}$
$25 + 1$
$d = 26$
midpoint formula
$(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$
example: find the midpoint between $(-6, 5)$ and $(8, 1)$
$\frac{-6 + 8}{2}$ $\frac{5 + 1}{2}$ $\frac{2}{2}$ $\frac{6}{2}$
$m = 4$
part 1: midpoint and distance formula practice
find the coordinates of the midpoint of the segment joining the given points.

1. $(0, 2)$ and $(6, 4)$.
2. $(-2, 2)$ and $(6, 4)$
3. $(6, -7)$ and $(-6, 3)$
4. $(-11, 3)$ and $(8, -7)$

find the distance between the two points.

1. $(-4, 2)$ and $(2, -1)$
2. $(-2, -3)$ and $(-2, 4)$
3. $(3, 2)$ and $(5, -2)$
4. $(5, -7)$ and $(8, -2)$

Explanation:

1. For mid - point of \((0,2)\) and \((6,4)\)

Step1: Apply mid - point formula for \(x\) - coordinate

The mid - point formula for \(x\) - coordinate is \(x=\frac{x_1 + x_2}{2}\). Here \(x_1 = 0,x_2=6\), so \(x=\frac{0 + 6}{2}=3\)

Step2: Apply mid - point formula for \(y\) - coordinate

The mid - point formula for \(y\) - coordinate is \(y=\frac{y_1 + y_2}{2}\). Here \(y_1 = 2,y_2 = 4\), so \(y=\frac{2+4}{2}=3\)

2. For mid - point of \((-2,2)\) and \((6,4)\)

Step1: Apply mid - point formula for \(x\) - coordinate

Using \(x=\frac{x_1 + x_2}{2}\), with \(x_1=-2,x_2 = 6\), we get \(x=\frac{-2 + 6}{2}=\frac{4}{2}=2\)

Step2: Apply mid - point formula for \(y\) - coordinate

Using \(y=\frac{y_1 + y_2}{2}\), with \(y_1 = 2,y_2=4\), we get \(y=\frac{2 + 4}{2}=3\)

3. For mid - point of \((6,-7)\) and \((-6,3)\)

Step1: Apply mid - point formula for \(x\) - coordinate

Using \(x=\frac{x_1 + x_2}{2}\), with \(x_1 = 6,x_2=-6\), we get \(x=\frac{6+( - 6)}{2}=\frac{0}{2}=0\)

Step2: Apply mid - point formula for \(y\) - coordinate

Using \(y=\frac{y_1 + y_2}{2}\), with \(y_1=-7,y_2 = 3\), we get \(y=\frac{-7 + 3}{2}=\frac{-4}{2}=-2\)

4. For mid - point of \((-11,3)\) and \((8,-7)\)

Step1: Apply mid - point formula for \(x\) - coordinate

Using \(x=\frac{x_1 + x_2}{2}\), with \(x_1=-11,x_2 = 8\), we get \(x=\frac{-11 + 8}{2}=\frac{-3}{2}=-1.5\)

Step2: Apply mid - point formula for \(y\) - coordinate

Using \(y=\frac{y_1 + y_2}{2}\), with \(y_1 = 3,y_2=-7\), we get \(y=\frac{3+( - 7)}{2}=\frac{-4}{2}=-2\)

7. For distance between \((-4,2)\) and \((2,-1)\)

Step1: Apply distance formula for \(x\) and \(y\) differences

The distance formula is \(D=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1=-4,x_2 = 2,y_1 = 2,y_2=-1\)
\(x_2 - x_1=2-( - 4)=6\) and \(y_2 - y_1=-1 - 2=-3\)

Step2: Calculate the distance

\(D=\sqrt{6^{2}+(-3)^{2}}=\sqrt{36 + 9}=\sqrt{45}=3\sqrt{5}\approx6.71\)

8. For distance between \((-2,-3)\) and \((-2,4)\)

Step1: Apply distance formula for \(x\) and \(y\) differences

Here \(x_1=-2,x_2=-2,y_1=-3,y_2 = 4\)
\(x_2 - x_1=-2-( - 2)=0\) and \(y_2 - y_1=4-( - 3)=7\)

Step2: Calculate the distance

\(D=\sqrt{0^{2}+7^{2}}=\sqrt{0 + 49}=7\)

9. For distance between \((3,2)\) and \((5,-2)\)

Step1: Apply distance formula for \(x\) and \(y\) differences

Here \(x_1 = 3,x_2=5,y_1 = 2,y_2=-2\)
\(x_2 - x_1=5 - 3=2\) and \(y_2 - y_1=-2 - 2=-4\)

Step2: Calculate the distance

\(D=\sqrt{2^{2}+(-4)^{2}}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\approx4.47\)

10. For distance between \((5,-7)\) and \((8,-2)\)

Step1: Apply distance formula for \(x\) and \(y\) differences

Here \(x_1 = 5,x_2=8,y_1=-7,y_2=-2\)
\(x_2 - x_1=8 - 5=3\) and \(y_2 - y_1=-2-( - 7)=5\)

Step2: Calculate the distance

\(D=\sqrt{3^{2}+5^{2}}=\sqrt{9 + 25}=\sqrt{34}\approx5.83\)

Answer:

1. \((3,3)\)
2. \((2,3)\)
3. \((0,-2)\)
4. \((-1.5,-2)\)
5. \(3\sqrt{5}\approx6.71\)
6. \(7\)
7. \(2\sqrt{5}\approx4.47\)
8. \(\sqrt{34}\approx5.83\)