Question

topic 3: distance & midpoint formula
distance formula: $sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}$

1. find $st$ if $s(-3,10)$ and $t(-2,3)$.
2. find $bc$ if $b(8,-7)$ and $c(-4,-2)$.
3. given the graph below, find $wv$.
4. given the graph below, find $pq$.
5. find the coordinates of the midpoint of $overline{hk}$ if $h(-1,2)$ and $k(-7,-4)$.
6. find the coordinates of $z$ if $y$ is the midpoint of $overline{xz}$, $x(-10,9)$, and $y(-4,8)$.

topic 4: angle measures

1. if $mangle def = 117^{circ}$, find the value of $x$.
2. if $mangle pqs = 16^{circ}$, $mangle sqr=(9x + 17)^{circ}$, and $mangle pqr=(12x - 6)^{circ}$, find $mangle pqr$.

Explanation:

14.

Step1: Identify distance - formula variables

Let \(S(x_1,y_1)=(-3,10)\) and \(T(x_2,y_2)=(-2,3)\). The distance formula is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Substitute values into formula

\(d=\sqrt{(-2-(-3))^2+(3 - 10)^2}=\sqrt{( - 2 + 3)^2+(3 - 10)^2}=\sqrt{1^2+( - 7)^2}=\sqrt{1 + 49}=\sqrt{50}=5\sqrt{2}\)

Step1: Identify distance - formula variables

Let \(B(x_1,y_1)=(8,-7)\) and \(C(x_2,y_2)=(-4,-2)\). The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Substitute values into formula

\(d=\sqrt{(-4 - 8)^2+(-2-(-7))^2}=\sqrt{(-12)^2+( - 2 + 7)^2}=\sqrt{144 + 25}=\sqrt{169}=13\)

Step1: Recall mid - point formula

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Let \(H(x_1,y_1)=(-1,2)\) and \(K(x_2,y_2)=(-7,-4)\).

Step2: Substitute values into formula

\(M=(\frac{-1+( - 7)}{2},\frac{2+( - 4)}{2})=(\frac{-8}{2},\frac{-2}{2})=(-4,-1)\)

Answer:

\(5\sqrt{2}\)

15.