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Drawing geometric figures on a coordinate plane allows to prove simple geometric theorems algebraically. In this lesson, several theorems will be proven algebraically.

### Catch-Up and Review

Here are a few recommended readings before getting started.

Try your knowledge on these topics.

a State the coordinates of the points plotted on the plane. b Match each concept with its corresponding diagram. c True or false?
A parallelogram has exactly two pairs of parallel sides.
In a parallelogram, adjacent angles are congruent.
The four sides of a parallelogram are congruent.
In a parallelogram, opposite sides and opposite angles are congruent.
The diagonals of a parallelogram bisect each other.
Adjacent angles in a parallelogram are supplementary.
d Use the Pythagorean Theorem to find the missing side length. ## Investigating the Properties of a Parallelogram

The figure below looks like a parallelogram. Without using measuring tools such as a ruler or a protractor, how can it be proven that the quadrilateral above is actually a parallelogram?

## Distance Formula

In certain situations, it is required to find the length of a line segment or a side of a polygon. To find those lengths, it is recommended to calculate the distance between the endpoints of the segment, or between two vertices of a polygon. If those points are plotted on a coordinate plane, the Distance Formula can be used.

Given two points and on a coordinate plane, their distance is given by the following formula.

### Proof

Start by plotting and on the coordinate plane. Both points can be arbitrarily plotted in Quadrant I for simplicity. Note that the position of the points in the plane does not affect the proof. Assume that is greater than and that is greater than Next, draw a right triangle. The hypotenuse of this triangle will be the segment that connects points and The difference between the coordinates of the points is the length of one of the legs of the triangle. Furthermore, the length of the other leg is given by the difference between the coordinates. Therefore, the lengths of the legs are and Now, consider the Pythagorean Equation.
Here, and are the lengths of the legs, and the length of the hypotenuse of a right triangle. Substitute the expressions for the legs for and to find the hypotenuse's length. Then, the equation can be solved for
Solve for
Note that, when solving for only the principal root was considered. The reason is that represents the length of a side and therefore must be positive. Keeping in mind that is the distance between and then By the Transitive Property of Equality, the Distance Formula is obtained.
In this lesson, the Distance Formula will be used to prove properties of geometric figures.

## Practice Finding the Distance Between Two Coordinate Pairs

Use the Distance Formula to calculate the distance between the points plotted in the coordinate plane. If needed, round the answer to two decimal places. ## Classifying a Parallelogram Using Its Coordinates

Ramsha and Davontay are learning to make picture frames so they want to determine whether the quadrilateral shown below is a rectangle. Ramsha says that is a rectangle. Davontay says it is not. Use the Distance Formula to decide who is correct.

### Hint

Other than just using the Distance Formula, try the Converse of the Pythagorean Theorem to determine whether the angles of the quadrilateral are right angles.

### Solution

If the four angles of the quadrilateral are right angles, then is a rectangle. First, it will be determined whether is a right angle. To do so, start by drawing the diagonal and considering By the Converse of the Pythagorean Theorem, if squared is equal to the sum of the squares of and then is a right triangle. In this case, then is a right angle. To find the coordinates of and will be substituted into the Distance Formula.
Evaluate right-hand side
The distance between and and therefore the length of the diagonal is The same procedure can be used to find and
Segment Points Substitute Simplify
Finally, as previously mentioned, it needs to be seen whether squared is equal to the sum of the squares of and
Evaluate
Since a true statement was obtained, it can be said that By the Converse of the Pythagorean Theorem, is a right triangle. Therefore, is a right angle. By following the same procedure, it can be shown that and are also right angles. Since each angle of is a right angle, the quadrilateral is a rectangle. Therefore, Ramsha is correct. She's on her way toward making some nice frames.

## Determining If a Point Is On a Circle's Circumference

Consider the circle centered at the origin and containing the point Maya has been asked to determine whether the point lies on the above circle. Use the Distance Formula to help Maya find the answer.

### Hint

Start by finding the radius of the circle.

### Solution

The radius of a circle is constant. Therefore, if lies on the circle, then its distance from the origin would also equal the radius of the circle. Point is given, so the radius can be solved by finding the distance between point and the origin, using the Distance Formula.
Evaluate right-hand side
The distance between the origin and and therefore the radius of the circle, is Using the Distance Formula one more time, the distance between the origin and will be calculated.
Evaluate right-hand side
The distance between and the origin is the same as the radius of the circle, Therefore, the point lies on the circle. ## Definition of a Midpoint

Sometimes the length of a segment is not what is needed. Instead, the coordinates of the point that lies exactly in the middle of a segment are needed.

## Midpoint

The midpoint of a line segment is the point that divides the segment into two segments of equal length. The point is the midpoint of the segment since the distance from to is the same as the distance from to

If the points are plotted on a coordinate plane, the Midpoint Formula can be used to find the coordinates of the midpoint.

## Midpoint Formula

The midpoint between two points and on a coordinate plane can be determined by the following formula.

The formula above is called the Midpoint Formula.

### Proof

For simplicity, the points and will be arbitrarily plotted in Quadrant I. Also, consider the line segment that connects these points. The midpoint between and is the midpoint of this segment. Note that the position of the points in the plane does not affect the proof. Consider the horizontal distance and the vertical distance between and . Since is the midpoint, splits each distance, and , in half. Therefore, the horizontal and vertical distances from each endpoint to the midpoint are and Let and be the coordinates of Now, focus on the coordinates. The difference between the corresponding coordinates gives the horizontal distances between the midpoint and the endpoints.
The graph above shows that these distances are both equal to Therefore, by the Transitive Property of Equality, they are equal.
This equation can be solved to find the coordinate of the midpoint
Solve for
The coordinate of is In the same way, it can be shown that the coordinate of is With this information, the coordinates of can be expressed in terms of the coordinates of and

The Midpoint Formula can also be used to prove some properties of geometric figures.

## Practice Finding Midpoints Using the Midpoint Formula

Use the Midpoint Formula to calculate the coordinates of the midpoint between the points plotted on the coordinate plane. ## Proving the Properties of a Parallelogram

Paulina has joined the bandwagon of making picture frames. She wants to prove that the diagonals of the parallelogram shown below bisect each other. Help Paulina become an awesome frame maker by proving the statement. 