Question

1. triangle abc has coordinates a (3, 9), b (5,1) and c (9, 5). d is the mid - point of ab and e is the mid - point of ac.

a) graph the points a, b, and c (make sure you label them). find the coordinates of points d and e. show all work.
d =
e =
b) plot points d and point e on the graph and label.
c) find the length of de. show all work.

Explanation:

Step1: Find mid - point formula

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

Step2: Calculate coordinates of D

For points $A(3,9)$ and $B(5,1)$, $x_D=\frac{3 + 5}{2}=\frac{8}{2}=4$, $y_D=\frac{9+1}{2}=\frac{10}{2}=5$. So $D=(4,5)$.

Step3: Calculate coordinates of E

For points $A(3,9)$ and $C(9,5)$, $x_E=\frac{3 + 9}{2}=\frac{12}{2}=6$, $y_E=\frac{9 + 5}{2}=\frac{14}{2}=7$. So $E=(6,7)$.

Step4: Calculate the length of DE

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, $x_1 = 4,y_1 = 5,x_2=6,y_2 = 7$. Then $d=\sqrt{(6 - 4)^2+(7 - 5)^2}=\sqrt{2^2+2^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$.

Answer:

a) $D=(4,5)$, $E=(6,7)$
b) (Graphing part is best done on paper or using a graphing tool. Plot $D(4,5)$ and $E(6,7)$ and label them.)
c) $2\sqrt{2}$