Question

three vertices of a rectangle are (0,5), (7,5), and (0,8). a. find the coordinates of the fourth vertex of the rectangle. b. find the perimeter, p, of the rectangle. c. find the area, a, of the rectangle. a. the fourth vertex of the rectangle is . (type an ordered pair.) b. p = c. a =

Explanation:

Step1: Determine the fourth - vertex coordinates

In a rectangle, opposite sides are parallel. Given vertices $(0,5)$, $(7,5)$, and $(0,8)$. The $x$ - coordinate of the fourth vertex will be the same as the $x$ - coordinate of $(7,5)$ and the $y$ - coordinate will be the same as the $y$ - coordinate of $(0,8)$. So the fourth vertex is $(7,8)$.

Step2: Calculate the length and width

The length $l$ of the rectangle is the distance between $(0,5)$ and $(7,5)$. Using the distance formula for points with the same $y$ - value $d=\vert x_2 - x_1\vert$, we have $l=\vert7 - 0\vert=7$. The width $w$ of the rectangle is the distance between $(0,5)$ and $(0,8)$. Using the distance formula for points with the same $x$ - value $d=\vert y_2 - y_1\vert$, we have $w=\vert8 - 5\vert = 3$.

Step3: Calculate the perimeter

The perimeter formula of a rectangle is $P = 2(l + w)$. Substituting $l = 7$ and $w = 3$ into the formula, we get $P=2(7 + 3)=2\times10 = 20$.

Step4: Calculate the area

The area formula of a rectangle is $A=l\times w$. Substituting $l = 7$ and $w = 3$ into the formula, we get $A=7\times3 = 21$.

Answer:

a. $(7,8)$
b. $20$
c. $21$