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This lesson will explore the use of coordinates to prove simple geometric theorems algebraically.

### Catch-Up and Review

Here is some recommended reading before getting started.

Try your knowledge on these topics.

a Find the standard equation of the circle. Write the answer in the form where are the coordinates of the center, and is the radius. b Consider lines and on the coordinate plane. Select all the statements that are true.

c Consider lines and on the coordinate plane. Select all the statements that are true.

## Investigating a Parallelogram in a Coordinate Plane

Without using the Pythagorean Theorem nor the Distance Formula, can it be determined whether the quadrilateral shown below is a rectangle? ## Proving Statements Using Different Methods

In mathematics, there are usually several ways of proving or disproving a statement. The following example will explore two ways of determining whether a point is on a circle.

Ramsha and Dylan are taking Geometry together. They were asked to determine whether the point is on the circle that passes through and whose center is at the origin. They decided to do it using different methods. Use both methods to determine whether the point is on the circle.

### Hint

The standard equation of the circle with center at and radius is The distance between two points with coordinates and is

### Solution

Ramsha has decided to write the standard equation of the circle, and then check whether the coordinates of satisfy this equation. Conversely, Dylan has decided to use the Distance Formula. These two options will be explored one at a time.

### Ramsha's Solution

As already said, Ramsha has decided first to write the standard equation of the circle and use it to see whether the point is on that circle. To do so, she will follow two steps.

1. Find the equation of the circle.
2. Verify whether the coordinates of the point satisfy the equation.

These steps will be done one at a time.

#### Finding the Equation of the Circle

The standard equation of a circle with center at and radius can be expressed.
Since the center is it is known that and Furthermore, since is on the circle, the distance between this point and the origin is the radius of the circle. This distance can be found by using the Distance Formula.
Simplify right-hand side
The radius of the circle is With this information, the standard equation of the circle can be written.
The circle with center at the origin and radius is shown below. #### Verifying If the Coordinates of the Point Satisfy the Equation

Finally, to determine whether point is on the circle, Ramsha will substitute and in the obtained equation. If a true statement is obtained, then the point is on the circle. Otherwise, the point is not on the circle.
Evaluate
A true statement was obtained. Therefore, Ramsha can conclude point is on the circle that passes through and whose center is the origin.

### Dylan's Solution

As previously stated, Dylan has decided to use the Distance Formula. To do so, he will follow two steps.

1. Find the radius of the circle.
2. Verify whether the distance from the given point to the center of the circle is the same as the radius.

These steps will be done one at a time.

#### Finding the Radius of the Circle

Since is the center of the circle and is a point on the circle, the distance between these points is the radius of the circle.
Simplify right-hand side
Dylan found that the radius of the circle is

#### Calculating the Distance From the Point to the Center of the Circle

If point is on the circle, then its distance from must also be Once again, the Distance Formula can be used.
Simplify right-hand side
The distance between and is equal to which is the same as the radius of the circle. Therefore, Dylan can conclude that the point is on the circle that passes through and whose center is the origin. ## Solving Real Life Problems Involving Triangles

Real life problems can also be solved by setting a coordinate plane and using geometric properties.

A small country has three main roads connecting the beach, the mountains, and a national park. To avoid traffic jams and car accidents during summer holidays, the government of the country is planning to build a fourth road to connect the midpoints of Road and Road Due to financial and logistic issues, building the fourth road would only be possible if it is parallel to and half the length of Road By placing the diagram on a coordinate plane, determine whether it is possible to build the new road.

### Hint

After placing the diagram on a coordinate plane, use the Midpoint Formula to find the coordinates of the midpoints of Road and Road

### Solution

First, the diagram will be placed on a coordinate plane. It can be seen in the above diagram that the beach, the mountains, and the national park are located at and respectively. To find the coordinates of the midpoints of Road and Road the Midpoint Formula can be used.
Midpoint Formula: Now that the coordinates of the midpoints of Road and Road are known, the length of the new road can be calculated. Also, the length of Road will be calculated to compare their lengths. These calculations can be done by using the Distance Formula.
Distance Formula:
Points Substitute Simplify
and
and
It can be seen in the above table that the distance between points and is half the distance between points and
Therefore, the new road would be half the length of Road Finally, the last step is to check whether this new road would be parallel to road To do so, the slope between points and will be compared to the slope between points and For this, the Slope Formula can be used.
Slope Formula:
Points Substitute Simplify
and
and
It was found that the slope between and is the same as the slope between and
By the Slopes of Parallel Lines Theorem, the new road would be parallel to Road In conclusion, if the new road connects the midpoints of Road and Road then it would be parallel to Road and half its length. Therefore, it is possible to construct.

## Triangle Midsegment Theorem

The result obtained in the previous example can be generalized to any triangle.

The line segment that connects the midpoints of two sides of a triangle — also known as a midsegment — is parallel to the third side of the triangle and half its length. If is a midsegment of then the following statement holds true.

and

### Proof

Using Coordinates

This theorem can be proven by placing the triangle on a coordinate plane. For simplicity, vertex will be placed at the origin and vertex on the axis. Since lies on the origin, its coordinates are Point is on the axis, meaning its coordinate is The remaining coordinates are unknown and can be named and
If is the midsegment from to then by the definition of a midpoint, and are the midpoints of and respectively. To prove this theorem, it must be proven that is parallel to and that is half of

If the slopes of these two segments are equal, then they are parallel. The coordinate of both and is Therefore, is a horizontal segment. Next, the coordinates of and will be found using the Midpoint Formula.

Segment Endpoints Substitute Simplify
and
and
The coordinate of both and is Therefore, is also a horizontal segment. Since all horizontal segments are parallel, it can be said that and are parallel.

Since both and are horizontal, their lengths are given by the difference of the coordinates of their endpoints.

Segment Endpoints Length Simplify
and
and
Since is half of it can be stated that the midsegment is half the length of
Therefore, a midsegment of two sides of a triangle is parallel to the third side of the triangle and half its length.

## Solving Real Life Problems Involving Rectangles

Not only can coordinates be used to prove properties of triangles, but also they can be used to prove properties of quadrilaterals.

Davontay lives in a neighborhood in Boston where the stadium, the amusement park, the airport, and the train station are the vertices of a quadrilateral that seems to be a parallelogram. Davontay knows that to determine whether a quadrilateral is a parallelogram, it is enough with to prove that opposite sides are congruent. Additionally, he wants to determine whether the diagonals bisect each other. Help Davontay to prove or deny these statements!

See solution.

### Hint

Place the diagram on a coordinate plane.

### Solution

First, the diagram will be placed on a coordinate plane. Davontay wants to see whether the opposite sides are congruent, and whether the diagonals bisect each other. These two things will be done one at a time.

### Opposite Sides are Congruent

In the previous diagram, it can be seen that the stadium, the amusement park, the airport, and the train station are at and respectively. These coordinates can be substituted into the Distance Formula to calculate the side lengths of the quadrilateral. The length of will be calculated first.
Evaluate right-hand side
By following the same procedure, all side lengths can be found.
Side Endpoints Substitute Simplify
and
and
and
and
By the Transitive Property of Equality, and are equal, and and are also equal.