Applying Coordinate Geometry

Standard

G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent.

Goals for this section:

Students should be able to use the x- and y-axis to describe coordinates for shapes. A coordinate grid can help to find the lengths of sides of shapes.

Useful formulas

Distance Formula

(x_1,y₁) is an endpoint.
(x_2,y₂) is a second endpoint.

This can help find the distance from one point to another and is especially useful for segments with a slope (it's not horizontal or vertical).

Midpoint Formula

(x_1,y₁) is an endpoint.
(x_2,y₂) is a second endpoint.

This can find what the middle point is between two other points, which is given as a coordinate.

Example

Given:

The height of the rectangle is a units.
The length of the rectangle is 2b units.
The y-axis bisects segments AB and DC.

Problem:

What are the coordinates of the vertices of ABCD?

A is (-b, a) because:

A goes to the left b units, hence the x-coordinate is -b
A goes up a units, hence the y-coordinate is a

B is (b, a) becase:

B goes to the right b units, hence the x-coordinate is b
B goes up a units, hence the y-coordinate is a.

What will the coordinates of C and D be?

Answers:

Example

Given:

Segments KM and KL are congruent.
N is the midpoint of KM.
O is the midpoint of KL.

Problem:

Show that segments LN and MO are congruent.

Solution:

Use the midpoint formula to find the coordinates of N and O.
Use the distance formula to find the length of segments NL and OM.

O is (a, b)

N is (-a, b)

Now find the length of LN and MO

L is (2a, 0)
N is (-a, b)
M is (-2a, 0)
O is (a, b)

Notice how both calculations come to the same answer. So therefore, LN and MO have the same length.

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