Topics
9. Connecting Algebra & Geometry with Coordinates
topic badge
United States of AmericaTennessee
Geometry

9.02 Perimeter and area in the coordinate plane

Lesson
Questions
Adaptive
Worksheet
What do you remember?
1

For the following polygons, using the correct notation for line segment lengths:

i
Write an algebraic expression for the perimeter of the polygons.
ii
Write an algebraic expression for the area of the polygons.

Do not perform any calculations.

a
-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
y
b
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
2

For the following triangles:

i
Calculate AB, as an exact value.
ii
Calculate BC, as an exact value.
iii
Calculate AC, as an exact value.
iv
Calculate the perimeter of \triangle ABC, rounding your answer to two decimal places if necessary.
a
-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
y
b
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
3

For the following polygons:

i
Calculate XY.
ii
Calculate YZ.
iii
Calculate the area of the polygon, rounding your answer to one decimal places if necessary.
a
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
b
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
Let's practice
4

For the following polygons:

i
Calculate the perimeter of the polygon, rounding your answer to two decimal places if necessary.
ii
Calculate the area of the polygon, rounding your answer to two decimal places if necessary.
a
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
b
-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
3
4
5
y
c
-1
1
2
3
4
5
6
7
x
-1
1
2
3
4
5
6
7
y
d
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
e
-6
-5
-4
-3
-2
-1
1
2
3
4
x
-6
-5
-4
-3
-2
-1
1
2
3
4
y
f
-14
-12
-10
-8
-6
-4
-2
2
4
6
8
x
-2
2
4
6
8
10
12
y
5

The points A\left(2, 1\right), B\left(7, 3\right) and C\left(7, - 5 \right) are the vertices of a triangle. Determine the area of the triangle.

6

The points A\left(-2, -3\right), B\left(3, 4\right), C\left(-4,9 \right) and D\left(-9, 2 \right) are the vertices of a square. Determine the area of the square.

7

The points A\left(-2, -3\right), B\left(2, 5\right), C\left(10, 9\right) and D\left(6, 1 \right) are the vertices of a rhombus. Determine the perimeter of the rhombus as simplified radical value.

Let's extend our thinking
8

The hexagon ABCDEF has all sides congruent and vertices at A\left(-1, 3\right), B\left(4, 3\right), C\left(7, -1\right),D\left(4,-5\right), E\left(-1, -5\right) and F\left(-4, -1 \right). Determine:

a
The perimeter of ABCDEF
b
The area of ABCDEF
9

Shantelle is putting up a fence around her chicken coop. She has posts at poles at A\left(0, 3\right), B\left(-1, -1\right), C\left(1, -3 \right) and D \left(6,-2 \right). Each unit on the grid represents 1 foot. She purchased 18 feet of fencing.

a
Zenaida tells Shantelle that there is not going to be enough fencing. Explain how Zenaida has drawn this conclusion.
b
The posts at B\left(-1, -1\right), C\left(1, -3 \right) and D\left(6,-2 \right) have all be put into the ground with concrete, so can not be moved, but the post at point A has not been put in the ground yet. Determine if she can move the post at point A so that she has enough fencing. If she can, give an example of a new location for point A. If not, explain why not.
10

Gabriella and Joel are asked to calculate the area of the square below. Their solutions are shown below. Compare and contrast their solutions.

If you were helping a friend with this question, which approach would you take? Explain.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

Gabriella's Solution

1\displaystyle XY\displaystyle =\displaystyle \sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} Distance formula
2\displaystyle XY\displaystyle =\displaystyle \sqrt{\left(0--4\right)^2+\left(4-0\right)^2} Substitution property of equality
3\displaystyle XY\displaystyle =\displaystyle \sqrt{\left(4\right)^2+\left(4\right)^2} Simplify
4\displaystyle XY\displaystyle =\displaystyle 4\sqrt{2} Simplify
5\displaystyle \text{Area}\displaystyle =\displaystyle XY^2 Area formula for square
6\displaystyle \text{Area}\displaystyle =\displaystyle \left(4\sqrt{2}\right)^2 Substitution property of equality
7\displaystyle \text{Area}\displaystyle =\displaystyle 32 Simplify

Joel's Solution

1\displaystyle \text{Area}\displaystyle =\displaystyle \dfrac{YW \times XZ}{2} Area formula for rhombus
2\displaystyle \text{Area}\displaystyle =\displaystyle \dfrac{8 \times 8}{2} Substitution property of equality
3\displaystyle \text{Area}\displaystyle =\displaystyle \dfrac{64}{2} Simplify
4\displaystyle \text{Area}\displaystyle =\displaystyle 32 Simplify
11

The area of the triangle with vertices A\left(3, 0\right), B\left(8, 2\right) and C\left(8, - 6 \right) has been calculated.

a
If the triangle is translated up 3 units, determine if the area will change. Explain your answer.
b
If only the vertex A\left(3, 0\right) is translated up 3 units, determine if the area will change. Explain your answer.
Sign up to access Worksheet
Get full access to our content with a Mathspace account
Book a Demo
Sign up for free

Outcomes

G.N.Q.A.1

Use units as a way to understand real-world problems.*

G.N.Q.A.1.A

Use appropriate quantities in formulas, converting units as necessary.

G.GPE.A.1

Use coordinates to justify geometric relationships algebraically and to solve problems.

G.MP1

Make sense of problems and persevere in solving them.

G.MP2

Reason abstractly and quantitatively.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP4

Model with mathematics.

G.MP5

Use appropriate tools strategically.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

G.MP8

Look for and express regularity in repeated reasoning.

What is Mathspace

About Mathspace