Distance Formula and Midpoint Formula: Algebraic Proofs in Geometry

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.

Coordinate plane
Basic definitions
Transformations
Congruence
Theorems about parallelograms
Pythagorean Theorem
Circles

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.

Challenge

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?

Discussion

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

SubstituteII

▼

Solve for

SqrtEqn

RearrangeEqn

Rearrange equation

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.

Pop Quiz

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.

Example

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.

SubstitutePoints

Substitute &

▼

Evaluate right-hand side

Subtract term

Calculate power

Add terms

Split into factors

Calculate root

The distance

between

and

and therefore the length of the diagonal

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

SubstituteValues

Substitute values

▼

Evaluate

Add terms

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.

Example

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.

SubstitutePoints

Substitute &

▼

Evaluate right-hand side

SubTerms

Subtract terms

CalcPow

Calculate power

IdPropAdd

Identity Property of Addition

CalcRoot

Calculate root

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.

SubstitutePoints

Substitute &

▼

Evaluate right-hand side

Subtract terms

Add terms

Calculate root

The distance between

and the origin is the same as the radius of the circle,

Therefore, the point

lies on the circle.

Discussion

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.

Concept

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

is the same as the distance from

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

Rule

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

AddEqn

CommutativePropAdd

Commutative Property of Addition

DivEqn

The

coordinate of

In the same way, it can be shown that the

coordinate of

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.

Pop Quiz

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.

Example

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.

Answer

See solution.

Hint

Use the Midpoint Formula to show that the diagonals intersect at their midpoint.

Solution

The diagonals bisect each other if and only if they intersect at their midpoint. Start by drawing the diagonals

and

Then, identify the coordinates of their point of intersection.

It is seen above that the point of intersection of

and

If this point is the midpoint of each diagonal, then the diagonals bisect each other. To prove that

is the midpoint of

the coordinates of the endpoints

and

can be substituted into the Midpoint Formula.

SubstitutePoints

Substitute &

▼

Evaluate

AddTerms

Add terms

AddNeg

MoveNegNumToFrac

Put minus sign in front of fraction

FracToDiv

The midpoint of the diagonal

is the point of intersection of the diagonals. By following the same procedure, it can be shown that the midpoint of

is also


Diagonal	Endpoints	Substitute	Simplify

The midpoint of both diagonals is the same as their point of intersection. Therefore, the diagonals bisect each other.

Illustration

Using Rigid Motions to Prove a Theorem

Any figure on a coordinate plane — located at any position — can be transformed through a combination of rigid motions to have a vertex at the origin and a consecutive vertex on the

axis. By this reasoning, when proving a theorem for a figure with a vertex at the origin and a consecutive vertex on the

axis, that theorem is valid for any figure with the same shape and size.

Try to place a vertex of the polygon at the origin and a consecutive vertex on the positive

Extra

How to Use the Applet

The polygon can be translated by clicking inside it and dragging it.
The polygon can be rotated around the center of rotation by clicking and dragging any of its vertices.
The center of rotation can be moved wherever needed.

Example

Proving the Triangle Midsegment Theorem by Rigid Motions

Paulina's best friend is obsessed with triangles. He has requested for Paulina to make him a triangular picture frame. Suppose Paulina can prove the Triangle Midsegment Theorem for the triangle below. She will better understand how to work with triangles and therefore make a better frame!

Help Paulina improve her frame making skills!

Answer

See solution.

Hint

Translate the triangle so that it has one vertex at the origin and a consecutive vertex on the positive axis. Then, consider the midpoints of and

Solution

Start by recalling the Triangle Midsegment Theorem.

Triangle Midsegment Theorem
The segment that connects the midpoints of two sides of a triangle — a midsegment — is parallel to the third side of the triangle and half its length.

To make the proof easier,

will be translated so that it has one vertex at the origin and a consecutive vertex on the positive

axis. For simplicity, vertex

will be located at the origin and

on the positive

axis. This is done by translating the triangle

unit to the left and

units down.

The vertices after the translation are

and

Now the Triangle Midsegment Theorem can be proven for this triangle. Let

and

be the midpoints of

and

respectively.

The coordinates of the midpoints of

and

can be found by using the Midpoint Formula.


Side	Endpoints	Substitute	Simplify
	and
	and

It can be seen above that both

and

have the same

coordinate

This means that

is a horizontal segment. Since

is on the

axis, it can be said that it is also a horizontal segment. Therefore, the midsegment is parallel to

Finally, it needs to be proven that

is half

To do this, the Distance Formula will be used.


Segment	Endpoints	Substitute	Simplify
	and
	and

Since

is half

the length of

is half the length of

By following the same procedure, it can be proven that the other two midsegments of

are parallel and half the length of the third side. The Triangle Midsegment Theorem has been proven. Paulina has just leveled up and her friend will get a great frame!

Closure

Proving the Properties of a Parallelogram

The topics covered in this lesson can now be applied to the challenge. The figure below looks like a parallelogram; without using measuring tools, how can it be proven that the quadrilateral is a parallelogram?

Answer

See solution.

Hint

Place the quadrilateral on a coordinate plane. Then, translate it so that it has a vertex at the origin and a consecutive vertex on the axis.

Solution

As advised, the quadrilateral will be placed on a coordinate plane. Then, it will be translated so that a vertex ends up at the origin and a consecutive vertex on the

axis. For simplicity, the vertices have been labeled.

It is sufficient to prove that

has one pair of congruent parallel sides to prove that

is a parallelogram. Note that the

coordinate of

and

Therefore,

is parallel to the

axis. Also,

and

lie on the

axis, so

and

are parallel.

To prove that the sides have the same length, the Distance Formula will be used.


Segment	Points	Substitute	Simplify

The length of the opposite sides and is This means that and are congruent.

The quadrilateral has been proven to have a pair of congruent parallel sides. Therefore, it is a parallelogram.

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Catch-Up and Review

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