Question

1. find the length between c(6, 3) and d(-5, 4).
2. a rectangle in the coordinate plane has vertices f(0, -2) and i(12, -7). find the area if fg is 3.
3. if i(-3,5), w(-3,-1), and t(4,1), then find the perimeter of △wit shown below.
4. plot d(3,-2) e(1, 4) f(-4, -2) in the coordinate plane. calculate the area of the triangle.
5. plot g(-4,4) e(-1, -3) f(2, -3) in the coordinate plane. calculate the area of the triangle.

g.gpe.7

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $C(6,3)$ and $D(-5,4)$, $x_1 = 6,y_1=3,x_2=-5,y_2 = 4$.
\[

$$\begin{align*} d&=\sqrt{(-5 - 6)^2+(4 - 3)^2}\\ &=\sqrt{(-11)^2+1^2}\\ &=\sqrt{121 + 1}\\ &=\sqrt{122} \end{align*}$$

\]

Step2: For rectangle area

Given vertices $F(0,-2)$ and $I(12,-7)$ of a rectangle and $FG = 3$. First, find the length of $FI$ using the distance formula. $x_1=0,y_1=-2,x_2 = 12,y_2=-7$.
\[

$$\begin{align*} FI&=\sqrt{(12 - 0)^2+(-7+2)^2}\\ &=\sqrt{12^2+(-5)^2}\\ &=\sqrt{144 + 25}\\ &=\sqrt{169}\\ &=13 \end{align*}$$

\]
The area of a rectangle $A=FG\times FI$, so $A = 3\times13=39$.

Step3: For triangle perimeter

For $\triangle WIT$ with $I(-3,5),W(-3,-1),T(4,1)$.
Find $IW$: Since $x$ - coordinates are the same ($x=-3$), $IW=\vert5+1\vert = 6$.
Find $WT$: $x_1=-3,y_1=-1,x_2 = 4,y_2=1$.
\[

$$\begin{align*} WT&=\sqrt{(4 + 3)^2+(1 + 1)^2}\\ &=\sqrt{7^2+2^2}\\ &=\sqrt{49+4}\\ &=\sqrt{53} \end{align*}$$

\]
Find $IT$: $x_1=-3,y_1=5,x_2 = 4,y_2=1$.
\[

$$\begin{align*} IT&=\sqrt{(4 + 3)^2+(1 - 5)^2}\\ &=\sqrt{7^2+(-4)^2}\\ &=\sqrt{49 + 16}\\ &=\sqrt{65} \end{align*}$$

\]
The perimeter $P=6+\sqrt{53}+\sqrt{65}$.

Step4: For triangle area (D, E, F)

For $\triangle DEF$ with $D(3,-2),E(1,4),F(-4,-2)$.
Base: The distance between $D(3,-2)$ and $F(-4,-2)$ (since $y$ - coordinates are the same), $b=\vert3+4\vert=7$.
Height: The distance from $E(1,4)$ to the line $y = - 2$, $h=\vert4 + 2\vert=6$.
The area of a triangle $A=\frac{1}{2}\times b\times h=\frac{1}{2}\times7\times6 = 21$.

Step5: For triangle area (G, E, F)

For $\triangle GEF$ with $G(-4,4),E(-1,-3),F(2,-3)$.
Base: The distance between $E(-1,-3)$ and $F(2,-3)$ (since $y$ - coordinates are the same), $b=\vert2 + 1\vert=3$.
Height: The distance from $G(-4,4)$ to the line $y=-3$, $h=\vert4+3\vert = 7$.
The area of a triangle $A=\frac{1}{2}\times b\times h=\frac{1}{2}\times3\times7=\frac{21}{2}$.

Answer:

1. The length between $C$ and $D$ is $\sqrt{122}$.
2. The area of the rectangle is $39$.
3. The perimeter of $\triangle WIT$ is $6+\sqrt{53}+\sqrt{65}$.
4. The area of $\triangle DEF$ is $21$.
5. The area of $\triangle GEF$ is $\frac{21}{2}$.