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Parallelograms in the Coordinate Plane

Prove geometric statements algebraically using points.

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Coordinate Proofs

Sherry just bought some walkie talkies that have a one mile range. To get to her friend Matt's house, Sherry rides her bike half a mile east and then three quarters of a mile north. If she gives Matt one of the walkie talkies, will she be able to talk to him on her walkie talkie from her house? Model this situation with an equation or inequality and a graph. Then, answer the question.

Coordinate Proofs

In a coordinate proof, you are proving geometric statements using algebra and the coordinate plane. Some examples of statements you might prove with a coordinate proof are:

  • Prove or disprove that the quadrilateral defined by the points \begin{align*}(2,4),(1,2),(5,1),(4,-1)\end{align*}(2,4),(1,2),(5,1),(4,1) is a parallelogram.
  • Prove or disprove that the point \begin{align*}\left ( 1,\sqrt{7} \right )\end{align*}(1,7) lies on the circle centered at the origin containing the point \begin{align*}(0,4)\end{align*}(0,4).
  • Prove or disprove that the quadrilateral defined by the points \begin{align*}(8,-4),(0,2),(-10,2),(-6,4)\end{align*}(8,4),(0,2),(10,2),(6,4) is a trapezoid.

In order to be successful with coordinate proofs, you need to first remember the definitions and properties of different shapes. This is because you need to know what must be shown in order to prove or disprove the statement. For example, to prove that a quadrilateral is a parallelogram, you would need to show that the opposite sides are parallel.

Next, you will need to figure out how you can use algebra to verify that a given shape has the necessary properties. The table below summarizes some of the common properties or facts you might want to show, and how to do it with algebra. Note that there is often more than one way to complete a coordinate proof!

I want to show that...

With the help of algebra I should...

The shape has congruent sides.

Use the distance formula or Pythagorean Theorem to find the length of the sides and show that they are equal.

The shape has right angles.

Show that the sides are perpendicular by finding the slopes of the lines (they should be opposite reciprocals).

The shape has parallel sides.

Show that the sides are parallel by finding the slopes of the lines (they should be equal).

A point is on a shape.

Find the equation(s) of the shape and verify that the point satisfies the equation.

 

Let's do a few proofs together.  

1. Prove or disprove that the quadrilateral defined by the points \begin{align*}(2,4),(1,2),(5,1),(4,-1)\end{align*}(2,4),(1,2),(5,1),(4,1) is a parallelogram.

You don't necessarily have to plot the points, but it helps to visualize. It also allows you to check whether or not it looks like a parallelogram.

This definitely looks like a parallelogram. To prove that it is a parallelogram, remember that the definition of a parallelogram is a quadrilateral with two pairs of parallel sides. Therefore, one way to prove it is a parallelogram is to verify that the opposite sides are parallel. From algebra, remember that two lines are parallel if they have the same slope. Find the slope of each side and label it on the picture.

The slopes of the opposite sides are equal. Therefore, the opposite sides are parallel. Therefore, the quadrilateral is a parallelogram. 

2. Prove or disprove that the point \begin{align*}\left (1,\sqrt{7} \right )\end{align*}(1,7) lies on the circle centered at the origin containing the point \begin{align*}(0,4)\end{align*}(0,4).

First, find the equation of the circle. The general equation of a circle is:

\begin{align*}(x-h)^2+(y-k)^2=r^2\end{align*}(xh)2+(yk)2=r2

where \begin{align*}(h,k)\end{align*}(h,k) is the center of the circle and \begin{align*}r\end{align*}r is the radius of the circle. This circle is centered at the origin, so \begin{align*}(h,k)=(0,0)\end{align*}(h,k)=(0,0). The circle contains the point \begin{align*}(0,4)\end{align*}(0,4). From \begin{align*}(0,0)\end{align*}(0,0) to \begin{align*}(0,4)\end{align*}(0,4) is a distance of 4 units, so the radius of the circle must be 4. The equation of the circle is:

\begin{align*}x^2+y^2=16\end{align*}x2+y2=16

Now, to prove or disprove that the point \begin{align*}\left (1,\sqrt{7} \right )\end{align*}(1,7) lies on the circle, see if the point satisfies the equation of the circle:

\begin{align*}(1)^2+ \left ( \sqrt{7} \right )^2 &=16(?)\\ 1+7 &=16(?)\\ 8 &\ne 16 \end{align*}(1)2+(7)21+78=16(?)=16(?)16

The point does not satisfy the equation. Therefore, the point does NOT lie on the circle.

3. Prove or disprove that the quadrilateral defined by the points \begin{align*}(8,1),(6,9),(4,0),(2,8)\end{align*}(8,1),(6,9),(4,0),(2,8) is a rectangle.

It helps to start by plotting the points.

This shape appears to be a rectangle. To prove that it is a rectangle, remember that the definition of a rectangle is a quadrilateral with four right angles. Therefore, to prove it is a rectangle you must verify that all angles are right angles. From algebra, remember that two lines meet at right angles if they are perpendicular, and two lines are perpendicular if they have opposite reciprocal slopes. Find the slope of each side and label it on the picture.

The slopes of each pair of adjacent sides are opposite reciprocals. Therefore, adjacent sides are perpendicular. Therefore, adjacent sides meet at right angles. Therefore, the quadrilateral is a rectangle. 

Examples

Example 1

Earlier, you were asked if Sherry will be able to talk to Matt on her walkie talkie from her house. 

If you imagine all the points within one mile of Sherry's house (in the range of the walkie talkies), you will have a circle. If you let Sherry's house be the origin, the set of points that Sherry could connect to with her walkie talkie is modeled by the inequality \begin{align*}x^2+y^2\le 1\end{align*}x2+y21.

If Sherry lives at the point \begin{align*}(0,0)\end{align*}(0,0), Matt lives at the point \begin{align*}(0.5,0.75)\end{align*}(0.5,0.75). To find if she will be able to communicate with Matt at Matt's house with the walkie talkies, test if the point \begin{align*}(0.5,0.75)\end{align*}(0.5,0.75) is within the shaded region. You can plot the point:

You can also verify that his house is within the range algebraically by testing to see if the point makes the inequality true.

\begin{align*}(0.5)^2+(0.75)^2 &\le 1(?)\\ 0.25+0.5625 &\le 1(?)\\ 0.8125 &\le 1\end{align*}(0.5)2+(0.75)20.25+0.56250.81251(?)1(?)1

You have now verified both algebraically and graphically that Matt's house is within the range of the walkie talkies.

Example 2

Prove or disprove that the quadrilateral defined by the points \begin{align*}(4,0)\end{align*}(4,0)\begin{align*}(5,3)\end{align*}(5,3)\begin{align*}(1,1)\end{align*}(1,1), \begin{align*}(2,4)\end{align*}(2,4) is a square.

First, plot the points. Below, the points have been labeled with letters to help with the identification of them later.

This shape appears to be a square. To prove that it is a square, remember that the definition of a square is a quadrilateral with four congruent sides and four right angles. Therefore, to prove it is a square you must:

FIRST: Find the lengths of all sides and verify that they are equal. You can use the distance formula or the Pythagorean Theorem to do this.

    • \begin{align*}AB=\sqrt{3^2+1^2}=\sqrt{10}\end{align*}AB=32+12=10
    • \begin{align*}BC=\sqrt{3^2+1^2}=\sqrt{10}\end{align*}BC=32+12=10
    • \begin{align*}CD=\sqrt{3^2+1^2}=\sqrt{10}\end{align*}CD=32+12=10
    • \begin{align*}DA=\sqrt{3^2+1^2}=\sqrt{10}\end{align*}DA=32+12=10

Because all four sides are the same length, all four sides are congruent.

SECOND: Find the slopes of all four sides and verify that adjacent sides have opposite reciprocal slopes and therefore are perpendicular, creating right angles.

    • \begin{align*}Slope \ of \ AB=-\frac{1}{3}\end{align*}Slope of AB=13
    • \begin{align*}Slope \ of \ BC=3\end{align*}Slope of BC=3
    • \begin{align*}Slope \ of \ CD=-\frac{1}{3}\end{align*}Slope of CD=13
    • \begin{align*}Slope \ of \ DA=3\end{align*}Slope of DA=3

Because all adjacent sides have opposite reciprocal slopes, adjacent sides are perpendicular. This means that adjacent sides meet at right angles and the shape has four right angles. 

The shape has four congruent sides and four right angles. Therefore, it is a square.

Example 3

Prove or disprove that the quadrilateral defined by the points \begin{align*}(5,-1)\end{align*}(5,1)\begin{align*}(6,3)\end{align*}(6,3)\begin{align*}(1,1)\end{align*}(1,1), \begin{align*}(2,5)\end{align*}(2,5) is a rhombus.

First, plot the points.

This shape appears to be a rhombus. To prove that it is a rhombus, remember that the definition of a rhombus is a quadrilateral with four congruent sides. Therefore, to prove it is a rhombus you must verify that all sides are the same length. You can use the distance formula or the Pythagorean Theorem to do this.

  • \begin{align*}AB=\sqrt{4^2+2^2}=\sqrt{18}=3\sqrt{2}\end{align*}AB=42+22=18=32
  • \begin{align*}BC=\sqrt{4^2+1^2}=\sqrt{17}\end{align*}BC=42+12=17
  • \begin{align*}CD=\sqrt{4^2+2^2}=\sqrt{18}=3\sqrt{2}\end{align*}CD=42+22=18=32
  • \begin{align*}DA=\sqrt{4^2+1^2}=\sqrt{17}\end{align*}

Even though the shape looked like a rhombus, its four sides are not actually congruent. Therefore, this is NOT a rhombus. 

Example 4

Prove or disprove that the quadrilateral defined by the points \begin{align*}(8,-4)\end{align*}\begin{align*}(0,2)\end{align*}\begin{align*}(-10,2)\end{align*}, \begin{align*}(-6,4)\end{align*} is a trapezoid.

It helps to start by plotting the points.

This shape appears to be a trapezoid. To prove that it is a trapezoid, remember that the definition of a trapezoid is a quadrilateral with exactly one pair of parallel sides. Therefore, to prove it is a trapezoid you must verify that one pair of sides is parallel. From algebra, remember that two lines are parallel if they have the same slope. Find the slope of each side and label it on the picture.

The slopes of exactly one pair of sides are equal. Therefore, exactly one pair of opposite sides is parallel. Therefore, the quadrilateral is a trapezoid. 

Review

1. Prove or disprove that the quadrilateral defined by the points \begin{align*}(5,3),(3,5),(3,1),(1,3)\end{align*} is a square.

2. Prove or disprove that the quadrilateral defined by the points \begin{align*}(7,6),(6,8),(2,3),(1,5)\end{align*} is a rectangle.

3. Prove or disprove that the quadrilateral defined by the points \begin{align*}(4,0),(0,3),(0,-3),(-4,0)\end{align*} is a rhombus.

4. Prove or disprove that the quadrilateral defined by the points \begin{align*}(6,4),(1,3),(3,2),(-2,1)\end{align*} is a parallelogram.

5. Prove or disprove that the quadrilateral defined by the points \begin{align*}(2,1),(0,4),(-6,-2),(-4,3) \end{align*} is a trapezoid.

6. Prove or disprove that the quadrilateral defined by the points \begin{align*}(5,3),(5,8),(2,2),(1,5)\end{align*} is a kite.

7. Prove or disprove that the quadrilateral defined by the points \begin{align*}(0,-2),(-1,1),(-3,-3),(-4,0)\end{align*} is a square.

8. Prove or disprove that the quadrilateral defined by the points \begin{align*}(7,2),(5,4),(4,1),(2,3) \end{align*} is a rhombus.

9. Prove or disprove that the quadrilateral defined by the points \begin{align*}(5,-1),(6,3),(1,1),(2,5) \end{align*} is a parallelogram.

10. Prove or disprove that the quadrilateral defined by the points \begin{align*}(6,2),(3,4),(4,-1),(1,1) \end{align*} is a rectangle.

11. Prove or disprove that the point \begin{align*}(2,4)\end{align*} lies on the exterior of the circle centered at the point \begin{align*}(1,2)\end{align*} that passes through the point \begin{align*}(3,3)\end{align*}.

12. Prove or disprove that the point \begin{align*}(-6,2)\end{align*} lies on the circle centered at the point \begin{align*}(-3,-1)\end{align*} that passes through the point \begin{align*}(0,2)\end{align*}.

13. Prove or disprove that the point \begin{align*}(4,-1)\end{align*} lies on the interior of the circle centered at the point \begin{align*}(3,2)\end{align*} that passes through the point \begin{align*}(1,-1)\end{align*}.

14. Prove or disprove that the point \begin{align*}\left ( 1,2\sqrt{2}\right )\end{align*} lies on the circle centered at the origin that passes through the point \begin{align*}(0,3)\end{align*}.

15. Prove or disprove that the point \begin{align*}(1,7.5)\end{align*} lies on the interior of the circle centered at the point \begin{align*}(1,6)\end{align*} that passes through the point \begin{align*}(2,5)\end{align*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.7. 

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Vocabulary

Distance Formula

The distance between two points (x_1, y_1) and (x_2, y_2) can be defined as d= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.

Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides.

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by a^2 + b^2 = c^2, where a and b are legs of the triangle and c is the hypotenuse of the triangle.

Rectangle

A rectangle is a quadrilateral with four right angles.

Rhombus

A rhombus is a quadrilateral with four congruent sides.

Square

A square is a polygon with four congruent sides and four right angles.

Coordinate Proof

A coordinate proof is a proof of a geometric statement using algebra.

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