Question

What is the area of a rectangle with vertices at ​ (−6, 3) ​, ​ (−3, 6) ​ , (1, 2) , and (−2, −1) ?

Answer 1

(1) the area of a rectangle with vertices at ​ (−6, 3) ​, ​ (−3, 6) ​ , (1, 2) , and (−2, −1)

To find area of rectangle we need to find the length and width

Length = distance between (−6, 3)  and (−2, −1)

Distance = $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

=  $\sqrt{(-2-(-6))^2 + (-1-3)^2}$

=  $\sqrt{(4)^2 + (-4)^2}$=$\sqrt{(16) + 16}$ =$\sqrt{32}$

width = Distance between (−6, 3) ​, ​ (−3, 6)

Distance = $\sqrt{(-3-(-6))^2 + (6-3)^2}$

=$\sqrt{3^2+3^2}= \sqrt{18}$

Area = Length * width = $\sqrt{32} *\sqrt{18} = \sqrt{576}= 24$

(2)  the area of a triangle with vertices at (0, −2) , ​ (8, −2) ​ , and ​ (9, 1) ​

Area of triangle = $\frac{1}{2} * base * height$

base  is the distance between (0,-2) and (8,-2)

Distance =  $\sqrt{(8-0)^2 + (-2-(-2))^2}$ = 8

To find out height we take two vertices (8,-2) and (9,1)

Height is the change in y values = 1- (-2) = 3

base = 8  and height = 3

So area of triangle = $\frac{1}{2} * 8 * 3 = 12$

(3) the perimeter of a polygon with vertices at (−2, 1) , ​ (−2, 4) ​, (2, 7) , ​ (6, 4) ​, and (6, 1) ​

To find perimeter we add  the length of all the sides

Distance between (−2, 1)  and (−2, 4) = $\sqrt{(-2+2)^2 + (4-1)^2}$= 3

Distance between(−2, 4) ​and (2, 7) = $\sqrt{(2+2)^2 + (7-4)^2}$= 5

Distance between (2, 7)  and (6, 4) = $\sqrt{(6 - 2)^2 + (4-7)^2}$= 5

Distance between (6, 4) ​ and (6, 1) ​= $\sqrt{(6 - 6)^2 + (1-4)^2}$= 3

Distance between  (6, 1) and (−2, 1) ​= $\sqrt{(-2-6)^2 + (1-1)^2}$= 8

Perimeter = 3 + 5 + 5 + 3 + 8 = 24

(4) four coordinates are (-7,-1) (-6,4) (3,-3) and (4,2)

Length = Distance between  (3,-3) and (4,2) ​= $\sqrt{(4-3)^2 + (2+3)^2}$= $\sqrt{26}$

Width = Distance between (-6,4) and (4,2) ​= $\sqrt{(4+6)^2 + (2-4)^2}$= $\sqrt{104}$

Perimeter = 2(lenght + width) = 2*( $\sqrt{26}$+$\sqrt{104}$ )

= 30.6 units

Answer 2

Part (1): $\boxed{24 \text{sq. units}}$

Part (2): $\boxed{12 \text{sq. units}}$

Part (3): $\boxed{24 \text{ units}}$

Part (4): $\boxed{\bf option c}$

Further explanation:  

Part (1):

The vertices of the rectangle is are $(-6,-3),(-3,6),(1,2)\ \text{and}\ (-2,-1)$.

Step 1:

First we calculate the length and the breadth of the rectangle.

The length and breadth of the rectangle can be calculated by the use of distance formula.

$\begin{aligned}b&=\sqrt{(-3-(-6))^{2}+(6-3)^{2}}\\&=\sqrt{3^{2}+3^{2}}\\&=\sqrt{18}\\&=3\sqrt{2}\end{aligned}$

$\begin{aligned}l&=\sqrt{(2-6)^{2}+(1-(-3))^{2}}\\&=\sqrt{16+16}\\&=\sqrt{32}\\&=4\sqrt{2}\end{aligned}$

The length of the rectangle is $4\sqrt{2}$ and the breadth of the rectangle is $3\sqrt{2}$.  

Step 2:

The area of the rectangle is calculated as,

$\begin{aligned}\text{Area}&=l\times b\\&=4\sqrt{2}\times 3\sqrt{2}\\&=12\times 2\\&=21\end{aligned}$  

Thus the area of the rectangle is $\bf 24\text{\bf square units}$.

Part (2):

The vertices of the triangle are $(0,-2),(8,-2)\text{ and}\ (9,1)$.

Step 1:

First we calculate the length of the sides of the triangle.

Use the distance formula to calculate the distance the length of the side of the triangle.

The length of the first side with the coordinates $(0,-2)$ and $(8,-2)$ is calculated as,

$\begin{aligned}a&=\sqrt{(-2-(-2))^{2}+(8-0)^{2}}\\&=\sqrt{64}\\&=8\end{aligned}$

The length of the second side with the coordinates $(9,1)$ and $(8,-2)$ is calculated as,

$\begin{aligned}b&=\sqrt{(1-(-2))^{2}+(9-8)^{2}}\\&=\sqrt{9+1}\\&=\sqrt{10}\end{aligned}$

The length of the third side with the coordinates $(9,1)$ and $(0,-2)$ is calculated as,

$\begin{aligned}c&=\sqrt{(1-(-2))^{2}+(9-0)^{2}}\\&=\sqrt{9+81}\\&=3\sqrt{10}\end{aligned}$  

The length of the sides of the triangle are $3\sqrt{10},\sqrt{10}$ and $8$.

Step 2:

Now, we calculate the semi perimeter of the triangle.

$\begin{aligned}s&=\dfrac{3\sqrt{10}+\sqrt{10}+8}{2}\\&=\dfrac{4\sqrt{10}+8}{2}\\&=2\sqrt{10}+4\end{aligned}$  

Step 3:

Use the heron’s formula to calculate the area of the triangle.

$\begin{aligned}\text{Area}&=\sqrt{s(s-a)(s-c)}\\&=\sqrt{(2\sqrt{10}+4)(2\sqrt{10}+4-8)(2\sqrt{10}+4-\sqrt{10})(2\sqrt{10}+4-3\sqrt{10})}\\&=\sqrt{(2\sqrt{10}+4)(2\sqrt{10}-4)(\sqrt{10}+4)(4-\sqrt{10})}\\&=\sqrt{((2\sqrt{10})^{2}-4^{2})(4^{2}-(\sqrt{10})^{2})}\\&=\sqrt{(40-16)(16-10)}\\&=\sqrt{144}\\&=12\end{aligned}$  

Thus the area of the triangle is $\boxed{\bf 12\text{\bf square units}}$.

Part (3):

The vertices of the polygon are $(-2,1),(-2,4),(2,7),(6,4)$ and $(6,1)$.

Step 1:

Use the distance formula to calculate the distance the length of the side of the polygon.

The length of the side of polygon is calculated as follows:

$\begin{aligned}a&=\sqrt{(4-1)^{2}+(-2-(-2))^{2}}\\&=\sqrt{9}\\&=3\end{aligned}$  

$\begin{aligned}b&=\sqrt{(2-(-2)^{2}+(7-4)^{2})}\\&=\sqrt{14+9}\\&=\sqrt{25}\\&=5\end{aligned}$

$\begin{aligned}c&=\sqrt{(4-7)^{2}+(6-2)^{2}}\\&=\sqrt{9+16}\\&=5\end{aligned}$  

$\begin{aligned}d&=\sqrt{(1-4)^{2}+(6-6)^{2}}\\&=\sqrt{9+0}\\&=3\end{aligned}$  

$\begin{aligned}e&=\sqrt{(1-1)^{2}+(6-(-2))^{2}\\&=\sqrt{0+64}\\&=8\end{aligned}$  

The length of the sides of the polygon are $3,5,5,3$ and $8$.

Step 2:

Perimeter of the polygon is the sum of all sides.

Therefore, the perimeter of the polygon is calculated as,

$\begin{aligned}p&=3+5+5+3+8\\&=24\ \text{units}\end{aligned}$    

The perimeter of the polygon is $24\ \text{units}$.

Part (4):

The vertices of the rectangle are $(-7,1),(-6,4),(3,-3)$ and $(4,2)$.

Step 1:

The length and breadth of the rectangle with coordinates $(-6,4)$ and $(4,2)$ is calculated as,

$\begin{aligned}l&=\sqrt{(4-(-6))^{2}+(2-4)^{2}}\\&=\sqrt{100+4}\\&=\sqrt{104}\end{aligned}$  

$\begin{aligned}b&=\sqrt{(2-(-3))^{2}+(4-3)^{2}}\\&=\sqrt{25+1}\\&=\sqrt{26}\end{aligned}$  

The length of the rectangle is $\sqrt{104}$ and the breadth of the rectangle is $\sqrt{}26$.

Step 2:  

The perimeter of the rectangle is calculated as,

$\begin{aligned}P&=2(l+b)\\&=2\times(\sqrt{104}+\sqrt{26})\\&=2\times(10.198+5.099)\\& \approx30.6\end{aligned}$    

Thus the perimeter of the rectangle is $30.59\text{ units}$.

Therefore, the correct option for part (4) is $\boxed{\bf \text{option c}}$.

Learn more:  

1. Learn more about the equation of the circle brainly.com/question/1506955

2. Learn more about the angles to define a term brainly.com/question/1953744

Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Mensuration

Keywords:  Perimeter, area triangle, rectangle, herons’s formula, distance formula, side, length, width, polygon, coordinate, vertices, algebraic expression, identity.