Applying Coordinate Geometry
Applying Coordinate Geometry
Standard
G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent.
Goals for this section:
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
Useful formulas
Distance Formula
Distance Formula
- (x1,y1) is an endpoint.
- (x2,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
Midpoint Formula
- (x1,y1) is an endpoint.
- (x2,y2) is a second endpoint.
This can find what the middle point is between two other points, which is given as a coordinate.
Example
Example
Given:
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:
Problem:
What are the coordinates of the vertices of ABCD?
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
Example
Given:
Given:
- Segments KM and KL are congruent.
- N is the midpoint of KM.
- O is the midpoint of KL.
Problem:
Problem:
Show that segments LN and MO are congruent.
Show that segments LN and MO are congruent.
Solution:
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)
O is (a, b)
N is (-a, b)
N is (-a, b)
Now find the length of LN and MO
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.
Notice how both calculations come to the same answer. So therefore, LN and MO have the same length.