Unlock Geometric Theorems Using Coordinates, Distance Formula, and Circle Equation

Challenge

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?

Example

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

P (1, 3)

is on the circle that passes through

Q (0, 2)

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 $(h, k)$ and radius $r$ is $(x - h)^{2} + (y - k)^{2} = r^{2} .$ The distance $d$ between two points with coordinates $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} .$

Solution

Ramsha has decided to write the standard equation of the circle, and then check whether the coordinates of $P$ 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.

Find the equation of the circle.
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

(h, k)

and radius

r

can be expressed.

(x - h)^{2} + (y - k)^{2} = r^{2}

Since the center is

O (0, 0),

it is known that

h = 0

and

k = 0 .

Furthermore, since

Q (2, 0)

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.

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

SubstituteValues

Substitute values

r = (0 - 2)^{2} + (0 - 0)^{2}

Simplify right-hand side

SubTerms

Subtract terms

r = (- 2)^{2} + 0^{2}

CalcPow

Calculate power

r = 4 + 0

IdPropAdd

\IdPropAdd

r = 4

CalcRoot

Calculate root

r = 2

The radius of the circle is

r = 2 .

With this information, the standard equation of the circle can be written.

(x - 0)^{2} + (y - 0)^{2} = 2^{2}

The circle with center at the origin and radius

2

is shown below.

Verifying If the Coordinates of the Point Satisfy the Equation

Finally, to determine whether point

P (1, 3)

is on the circle, Ramsha will substitute

x = 1

and

y = 3

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.

(x - 0)^{2} + (y - 0)^{2} = 2^{2}

SubstituteII

$x = 1$ , $y = 3$

(1 - 0)^{2} + (3 - 0)^{2} = ? 2^{2}

Evaluate

SubTerms

Subtract terms

1^{2} + (3)^{2} = ? 2^{2}

CalcPow

Calculate power

1 + (3)^{2} = ? 4

PowSqrt

$(a)^{2} = a$

1 + 3 = ? 4

AddTerms

Add terms

4 = 4 ✓

A true statement was obtained. Therefore, Ramsha can conclude point

P (1, 3)

is on the circle that passes through

Q (0, 2)

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.

Find the radius of the circle.
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

O (0, 0)

is the center of the circle and

Q (2, 0)

is a point on the circle, the distance between these points is the radius

r

of the circle.

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

SubstituteValues

Substitute values

r = (0 - 2)^{2} + (0 - 0)^{2}

Simplify right-hand side

SubTerms

Subtract terms

r = (- 2)^{2} + 0^{2}

CalcPow

Calculate power

r = 4 + 0

IdPropAdd

\IdPropAdd

r = 4

CalcRoot

Calculate root

r = 2

Dylan found that the radius of the circle is

2 .

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

If point

P (1, 3)

is on the circle, then its distance from

O (0, 0)

must also be

2 .

Once again, the Distance Formula can be used.

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

SubstituteValues

Substitute values

2 = ? (0 - 1)^{2} + (0 - 3)^{2}

Simplify right-hand side

SubTerms

Subtract terms

2 = ? (- 1)^{2} + (- 3)^{2}

NegBaseToPosPow

$(- a)^{2} = a^{2}$

2 = ? 1^{2} + (3)^{2}

BaseOne

$1^{a} = 1$

2 = ? 1 + (3)^{2}

PowSqrt

$(a)^{2} = a$

2 = ? 1 + 3

AddTerms

Add terms

2 = ? 4

CalcRoot

Calculate root

2 = 2 ✓

The distance between

O (0, 0)

and

P (1, 3)

is equal to

2,

which is the same as the radius of the circle. Therefore, Dylan can conclude that the point

P (1, 3)

is on the circle that passes through

Q (0, 2)

and whose center is the origin.

Example

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 $1$ and Road $2 .$

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

3 .

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 $1$ and Road $2 .$

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

B (4, 0),

M (- 4, 4),

and

P (- 2, - 2),

respectively. To find the coordinates of the midpoints of Road

1

and Road

2,

the Midpoint Formula can be used.

Midpoint Formula: $(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Road	Points	Substitute	Simplify
Road $1$	$B (4, 0)$ and $M (- 4, 4)$	$M_{1} (\frac{4 + ( - 4 )}{2}, \frac{0 + 4}{2})$	$M_{1} (0, 2)$
Road $2$	$M (- 4, 4)$ and $P (- 2, - 2)$	$M_{2} (\frac{- 4 + ( - 2 )}{2}, \frac{4 + ( - 2 )}{2})$	$M_{2} (- 3, 1)$

The midpoint of Road

1

M_{1} (0, 2)

and the midpoint of Road

2

M_{2} (- 3, 1) .

Now that the coordinates of the midpoints of Road

1

and Road

2

are known, the length of the new road can be calculated. Also, the length of Road

3

will be calculated to compare their lengths. These calculations can be done by using the Distance Formula.

Distance Formula: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Points	Substitute	Simplify
$M_{1} (0, 2)$ and $M_{2} (- 3, 1)$	$d_{M_{1} M_{2}} = (- 3 - 0)^{2} + (1 - 2)^{2}$	$d_{M_{1} M_{2}} = 10$
$B (4, 0)$ and $P (- 2, - 2)$	$d_{B P} = (- 2 - 4)^{2} + (- 2 - 0)^{2}$	$d_{B P} = 210$

It can be seen in the above table that the distance between points

M_{1}

and

M_{2}

is half the distance between points

B

and

P .

10 = \frac{1}{2} (210) ⇓ d_{M_{1} M_{2}} = \frac{1}{2} d_{B P}

Therefore, the new road would be half the length of Road

3 .

Finally, the last step is to check whether this new road would be parallel to road

3 .

To do so, the slope between points

M_{1}

and

M_{2}

will be compared to the slope between points

B

and

P .

For this, the Slope Formula can be used.

Slope Formula: $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Points	Substitute	Simplify
$M_{1} (0, 2)$ and $M_{2} (- 3, 1)$	$m_{M_{1} M_{2}} = \frac{1 - 2}{- 3 - 0}$	$m_{M_{1} M_{2}} = \frac{1}{3}$
$B (4, 0)$ and $P (- 2, - 2)$	$m_{B P} = \frac{- 2 - 0}{- 2 - 4}$	$m_{B P} = \frac{1}{3}$

It was found that the slope between

M_{1}

and

M_{2}

is the same as the slope between

B

and

P .

\frac{1}{3} = \frac{1}{3} ⇓ m_{M_{1} M_{2}} = m_{B P}

By the Slopes of Parallel Lines Theorem, the new road would be parallel to Road

3 .

In conclusion, if the new road connects the midpoints of Road

1

and Road

2,

then it would be parallel to Road

3

and half its length. Therefore, it is possible to construct.

Discussion

Triangle Midsegment Theorem

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

Example

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!

Answer

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

S (- 2, - 3),

P (- 5, 4),

A (3, 5),

and

T (6, - 2),

respectively. These coordinates can be substituted into the Distance Formula to calculate the side lengths of the quadrilateral. The length of

\overline{S P}

will be calculated first.

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

SubstituteValues

Substitute values

S P = (- 5 - (- 2))^{2} + (4 - (- 3))^{2}

Evaluate right-hand side

SubNeg

$a - (- b) = a + b$

S P = (- 3)^{2} + 7^{2}

CalcPow

Calculate power

S P = 9 + 49

AddTerms

Add terms

S P = 58

By following the same procedure, all side lengths can be found.

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Side	Endpoints	Substitute	Simplify
$\overline{S P}$	$S (- 2, - 3)$ and $P (- 5, 4)$	$S P =$ $(- 5 - (- 2))^{2} + (4 - (- 3))^{2}$	$S P = 58$
$\overline{P A}$	$P (- 5, 4)$ and $A (3, 5)$	$P A =$ $(3 - (- 5))^{2} + (5 - 4)^{2}$	$P A = 69$
$\overline{A T}$	$A (3, 5)$ and $T (6, - 2)$	$A T =$ $(6 - 3)^{2} + (- 2 - 5)^{2}$	$A T = 58$
$\overline{T S}$	$T (6, - 2)$ and $S (- 2, - 3)$	$T S =$ $(- 2 - 6)^{2} + (- 3 - (- 2))^{2}$	$T S = 69$

By the Transitive Property of Equality,

S P

and

A T

are equal, and

P A

and

T S

are also equal.

{S P = 58 A T = 58 ⇓ S P = A T {P A = 69 T S = 69 ⇓ P A = T S

Finally, if two segments have the same length, then they are congruent. Therefore, it can be said that the opposite sides of the quadrilateral are congruent.

\overline{S P} ≅ \overline{A T} and \overline{P A} ≅ \overline{T S} ✓

This confirms that the quadrilateral is a parallelogram.

Diagonals Bisect Each Other

The diagonals of the parallelogram are the line segments that connect opposite vertices. In the diagram below, the diagonals will be drawn and their point of intersection

I

will be plotted.

The midpoint of a line segment is the point that divides the segment into two congruent segments. Therefore, if the diagonals intersect at their midpoint, then they bisect each other. To determine whether

I

is the midpoint of the diagonals, the Midpoint Formula can be used. The midpoint of

\overline{P T}

will be calculated first.

M_{1} (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

SubstitutePoints

Substitute $(- 5, 4)$ & $(6, - 2)$

M_{1} (\frac{- 5 + 6}{2}, \frac{4 + ( - 2 )}{2})

Evaluate

AddNeg

$a + (- b) = a - b$

M_{1} (\frac{- 5 + 6}{2}, \frac{4 - 2}{2})

AddSubTerms

Add and subtract terms

M_{1} (\frac{1}{2}, \frac{2}{2})

QuotOne

$\frac{a}{a} = 1$

M_{1} (\frac{1}{2}, 1)

The midpoint of the diagonal

\overline{P T}

M_{1} (\frac{1}{2}, 1) .

By following the same procedure, the midpoint of the diagonal

\overline{S A}

can be calculated.

$M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Diagonal	Endpoints	Substitute	Simplify
$\overline{P T}$	$P (- 5, 4)$ and $T (6, - 2)$	$M_{1} (\frac{- 5 + 6}{2}, \frac{4 + ( - 2 )}{2})$	$M_{1} (\frac{1}{2}, 1)$
$\overline{S A}$	$S (- 2, - 3)$ and $A (3, 5)$	$M_{2} (\frac{- 2 + 3}{2}, \frac{- 3 + 5}{2})$	$M_{2} (\frac{1}{2}, 1)$

It can be seen that the midpoint of each diagonal has coordinates

(\frac{1}{2}, 1) .

Therefore, they intersect at their midpoint.

It can be stated that the diagonals bisect each other.

\overline{P I} ≅ \overline{I T} and \overline{S I} ≅ \overline{I A} ✓

Discussion

Properties of Parallelograms

The results of the previous example can be generalized to any parallelogram.

Closure

Classifying a Parallelogram in a Coordinate Plane

The challenge presented at the beginning of this lesson can be solved by using coordinates.

Using neither the Pythagorean Theorem nor the Distance Formula, it is desired to determine whether the quadrilateral below is a rectangle.

Hint

By the Slopes of Perpendicular Lines Theorem, if the slopes of two lines or segments are opposite reciprocals, then lines are perpendicular.

Solution

For simplicity, the vertices will be labeled $A,$ $B,$ $C,$ and $D .$

Next, the slopes of the sides will be found by using the Slope Formula. The slope of

\overline{A B}

will be calculated first.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstituteValues

Substitute values

m_{\overline{A B}} = \frac{6 - 2}{- 3 - ( - 6 )}

Evaluate right-hand side

SubNeg

$a - (- b) = a + b$

m_{\overline{A B}} = \frac{6 - 2}{- 3 + 6}

AddSubTerms

Add and subtract terms

m_{\overline{A B}} = \frac{4}{3}

By following the same procedure, the slopes of the remaining sides can be found.

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Side	Endpoints	Substitute	Simplify
$\overline{A B}$	$A (- 6, 2)$ and $B (- 3, 6)$	$m_{\overline{A B}} = \frac{6 - 2}{- 3 - ( - 6 )}$	$m_{\overline{A B}} = \frac{4}{3}$
$\overline{B C}$	$B (- 3, 6)$ and $C (9, - 3)$	$m_{\overline{B C}} = \frac{- 3 - 6}{9 - ( - 3 )}$	$m_{\overline{B C}} = - \frac{3}{4}$
$\overline{D C}$	$D (6, - 7)$ and $C (9, - 3)$	$m_{\overline{D C}} = \frac{- 3 - ( - 7 )}{9 - 6}$	$m_{\overline{D C}} = \frac{4}{3}$
$\overline{A D}$	$A (- 6, 2)$ and $D (6, - 7)$	$m_{\overline{A D}} = \frac{- 7 - 2}{6 - ( - 6 )}$	$m_{\overline{A D}} = - \frac{3}{4}$

By the Slopes of Perpendicular Lines Theorem, if the slopes of two line segments are opposite reciprocals, then the segments are perpendicular. The slopes of consecutive sides will be multiplied to see if they are opposite reciprocals. Recall that, in a polygon, two sides are consecutive if they share a common vertex.

Sides	Product of Slopes
$\overline{A B}$ and $\overline{B C}$	$\frac{4}{3} (- \frac{3}{4}) = - 1 ✓$
$\overline{B C}$ and $\overline{D C}$	$- \frac{3}{4} (\frac{4}{3}) = - 1 ✓$
$\overline{D C}$ and $\overline{A D}$	$\frac{4}{3} (- \frac{3}{4}) = - 1 ✓$
$\overline{A D}$ and $\overline{A B}$	$- \frac{3}{4} (\frac{4}{3}) = - 1 ✓$

Since the product of their slopes is $- 1,$ the quadrilateral's consecutive sides are perpendicular. Therefore, they form right angles. Consequently, by its definition, the quadrilateral is a rectangle.

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Hint

Solution

Ramsha's Solution

Finding the Equation of the Circle

Verifying If the Coordinates of the Point Satisfy the Equation

Dylan's Solution

Finding the Radius of the Circle

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

Hint

Solution

Answer

Hint

Solution

Opposite Sides are Congruent

Diagonals Bisect Each Other

Hint

Solution