Question

1. the midpoint of $overline{ab}$ is $m(-2, -5)$. if the coordinates of $a$ are $(1, -8)$, what are the coordinates of $b$?
2. graph a right triangle with the points $(3, 3)$ and $(8, -9)$ forming the hypotenuse. using the sides, find the distance between the two points in simplest radical form.

leg 1: __ leg 2: hypotenuse: __

1. graph a right triangle with the points $(-5, -9)$ and $(-7, -2)$ forming the hypotenuse. using the sides, find the distance between the two points in simplest radical form.

leg 1: __ leg 2: hypotenuse: __
the midpoint of $overline{ab}$ is $m(-3, 1)$. if the coordinates of $a$ are $(1, 6)$, what are the coordinates of $b$?

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step2: Find the x - coordinate of B for the first problem

Given $M(-2,-5)$ and $A(1,-8)$. Let the coordinates of $B$ be $(x,y)$. For the x - coordinate, $\frac{1 + x}{2}=-2$. Cross - multiply: $1 + x=-4$. Then $x=-4 - 1=-5$.

Step3: Find the y - coordinate of B for the first problem

For the y - coordinate, $\frac{-8 + y}{2}=-5$. Cross - multiply: $-8 + y=-10$. Then $y=-10 + 8=-2$. So the coordinates of $B$ are $(-5,-2)$.

Step4: Recall distance formula for the second problem (points $(3,3)$ and $(8,-9)$)

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $x_1 = 3,y_1 = 3,x_2 = 8,y_2=-9$.

Step5: Calculate the differences

$x_2 - x_1=8 - 3 = 5$ and $y_2 - y_1=-9 - 3=-12$.

Step6: Calculate the distance

$d=\sqrt{5^2+(-12)^2}=\sqrt{25 + 144}=\sqrt{169}=13$.

Step7: Recall distance formula for the third problem (points $(-5,-9)$ and $(-7,-2)$)

Let $x_1=-5,y_1=-9,x_2=-7,y_2=-2$.

Step8: Calculate the differences

$x_2 - x_1=-7-(-5)=-2$ and $y_2 - y_1=-2-(-9)=7$.

Step9: Calculate the distance

$d=\sqrt{(-2)^2+7^2}=\sqrt{4 + 49}=\sqrt{53}$.

Answer:

For the first problem (finding B given mid - point and A): $(-5,-2)$
For the second problem (distance between $(3,3)$ and $(8,-9)$): $13$
For the third problem (distance between $(-5,-9)$ and $(-7,-2)$): $\sqrt{53}$