Perimeter and Area in a Coordinate Plane

The area and perimeter of a rectangle can be calculated using the following formulas. Area:& A=wl Perimeter:& P=2(w+l) Here, w and l are the width and the length of the rectangle. In the diagram, the coordinates of the vertices are given. Using the Distance Formula, the length and width of a rectangle can be calculated. d=sqrt((x_2-x_1)^2+(y_2-y_1)^2) First, the length and width of the rectangle need to be identified.

Next, by substituting the endpoints of the chosen segments into the Distance Formula, their lengths can be calculated.

	Length	Width
Endpoints	( - 5, 2) and ( 3, 5)	( 3, 5) and ( 5, 0)
Substitute	l=sqrt(( 3-( - 5))^2+( 5- 2)^2)	w=sqrt(( 5- 3)^2+( 0- 5)^2)
Evaluate	l≈ 8.5	w≈ 5.4

Now that the length and the width are known, the area and the perimeter of the rectangle can be determined.

	Area	Perimeter
Formula	A=wl	P=2(w+l )
Substitute	A= 5.4( 8.5)	P=2( 5.4+ 8.5)
Evaluate	A=45.9	P=27.8

Since these values are greater than the area and the perimeter of the plot of land that Izabella wanted to buy, they satisfy the given requirements. Therefore, Izabella should buy this plot of land.

First, the coordinates of the vertices on the coordinate plane need to be noted.

Next, the side lengths of △ MNK need to be calculated by using the Distance Formula. The calculations for MN are shown below.

By following the same procedure, NK and MK can be calculated.

d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Side	Endpoints	Substitute	Evaluate
MN	M( - 2, - 2) and N( 3, 4)	MN=sqrt(( 3-( - 2))^2+( 4-( - 2))^2)	MN≈ 7.8
NK	N( 3, 4) and K( 4, - 2)	NK=sqrt(( 4- 3)^2+( - 2- 4)^2)	NK≈ 6.1
MK	M( - 2, - 2) and K( 4, - 2)	MK=sqrt(( 4-( - 2))^2+( - 2-( - 2))^2)	MK=6

Now, the perimeter of △ MNK can be calculated by adding the three side lengths of the triangle. P=7.8+6.1+6 ⇕ P=19.9 The perimeter of △ MNK is 19.9 units. Next, its area will be found. Recall that the area of a triangle is half the product of the base and its corresponding height. A=bh/2 In this formula, b is the base of the triangle and h its corresponding height. In a triangle, the base and its corresponding height are perpendicular. If MK is the base, then the height is the perpendicular segment to MK through N. In the diagram, it can be seen that the length of this segment is 6 units.

Finally, by substituting b= 6 and h= 6, the area of △ MNK can be calculated.

Therefore, the area of the triangle is 18 square units and its perimeter 19.9 units. This means that Kriz was not correct.

Start by identifying the coordinates of the vertices and the lengths of the horizontal and vertical sides of the figure. Let m be the length of the side which is neither vertical nor horizontal.

The value of m can be calculated by substituting (- 5,1) and (0,4) into the Distance Formula.

By adding the lengths of the exterior sides, the perimeter of the plan can be calculated. P=3+5.8+6+6+6+5 ⇕ P=31.8 The perimeter of the figure is 31.8 units. The area of a compound figure is equal to the sum of the geometric figures' areas that make the compound figure. From the diagram, it can be seen that the flat has a square, a rectangular, and a triangular room.

Square Room

The area of a square is equal to the square of a side length. It has been previously found that, in the plan, each side of the square room is 6 units long.

By raising 6 to the power of two, the area of the square can be determined. \begin{gathered} A_\text{square}=6^2\\ \Updownarrow \\ A_\text{square}=36\text{ units}^2 \end{gathered}

Rectangular Room

The area of a rectangle is equal to the product of its length and width. From the diagram, it can be seen that the length of the rectangle is 5 units and the width is 3 units.

By multiplying 3 by 5, the area of the rectangle can be found. \begin{gathered} A_\text{rectangle}=3(5)\\ \Updownarrow \\ A_\text{rectangle}=15\text{ units}^2 \end{gathered}

Triangular Room

The area of a triangle is half the product of its base and its height. Analyzing the diagram, it can be seen that the base and height of the triangle are b=5 units and h=3 units, respectively.

By substituting these values into the formula for the area of a triangle, its area can be calculated.

Total Area

Gather all the areas that have been found. A_(square)&=36units^2 A_(rectangle)&=15units^2 A_(triangle)&=7.5units^2 By adding these values, the area of the compound figure can be calculated. \begin{gathered} A_\text{total}=36+15+7.5\\ \Updownarrow\\ A_\text{total}=58.5\text{ units}^2 \end{gathered}

To calculate the area of a triangle, Heron's formula can be used. A=sqrt(s(s-a)(s-b)(s-c)) Here, a, b, and c are the side lengths and s is the semi-perimeter of the triangle. The semi-perimeter is half the perimeter. s=a+b+c/2 In the diagram, the coordinates of the vertices are given. The sides of the triangle will be labeled as a, b, and c.

By substituting the coordinates of the vertices into the Distance Formula, the side lengths can be calculated.

d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Side	Endpoints	Substitute	Evaluate
a	( 2, 3) and ( 3, 7.5)	a=sqrt(( 3- 2)^2+( 7.5- 3)^2)	a≈ 4.6
b	( 3, 7.5) and ( 7, 1)	b=sqrt(( 7- 3)^2+( 1- 7.5)^2)	b≈ 7.6
c	( 2, 3) and ( 7, 1)	c=sqrt(( 7- 2)^2+( 1- 3)^2)	c≈ 5.4

Next, the side lengths will be substituted into the formula for the semi-perimeter to find the value of s.

Now that the semi-perimeter of the triangle is known, its area can be found using Heron's Formula.

The area of the triangle is approximately 12.3 square units.