Question

read the questions below carefully. clearly show all work, including annotations, appropriate formula substitutions, rams, units, and complete sentences when necessary. box all final answers. all work must be shown in order to receive full credit. **#6.) the length of a rectangle is x ft., the width is (x + 4)ft., and the area is 32 ft.². determine the measure of the width. (#/4) 7.) △abc is an equilateral triangle (all sides are congruent) and a(4, - 3) and c(16, 2) are the coordinates of one side. after determining the length of side ac, what is the perimeter of △abc?

Explanation:

Step1: Set up area formula for rectangle

The area formula for a rectangle is $A = lw$, where $l$ is length and $w$ is width. Given $l=x$, $w = x + 4$ and $A=32$, we substitute to get $x(x + 4)=32$.
\[x(x + 4)=32\Rightarrow x^{2}+4x-32 = 0\]

Step2: Factor the quadratic equation

We factor $x^{2}+4x - 32$ as $(x + 8)(x - 4)=0$.

Step3: Solve for $x$

Set each factor equal to zero:
If $x+8=0$, then $x=-8$. But length cannot be negative, so we discard this solution.
If $x - 4=0$, then $x = 4$.

Step4: Find the width

The width is $x + 4$. Substitute $x = 4$ into $x+4$, we get $4 + 4=8$ ft.

Step5: Calculate side - length of equilateral triangle

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 $A(4,-3)$ and $C(16,2)$, we have:
\[AC=\sqrt{(16 - 4)^{2}+(2+3)^{2}}=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13\]

Step6: Calculate perimeter of equilateral triangle

Since an equilateral triangle has all sides equal and side - length $s = 13$, the perimeter $P=3s$. So $P = 3\times13=39$.

Answer:

The width of the rectangle is 8 ft and the perimeter of $\triangle ABC$ is 39.