Distance

Essential Question: How do you use the distance formula to find distances and

lengths in the coordinate plane?

Distance in the Coordinate Plane

• You can use the Pythagorean Theorem to help you find the Distance between two points.

Distance in the Coordinate Plane

A(2,5) B(-4,-3)

a. Plot the points A and B in the coordinate plane on the right.

b. Draw c. Draw a vertical line through

point Ad. Draw a horizontal line through

point Be. Label the intersection of the

vertical and horizontal line as point C.

Distance in the Coordinate Plane

A(2,5) B(-4,-3)

• Each small square grid square is 1 unit by 1 unit. Use this fact to find the lengths of AC and BC

• AC = _______• BC = _______

Distance in the Coordinate Plane

A(2,5) B(-4,-3)

Pythagorean Theorem:

Distance in the Coordinate Plane

A(2,5) B(-4,-3)

Pythagorean Theorem:

Solve for AB.

Reflect 1a.

• Explain how you solved for AB in Step F.

Reflect 1b.

• Can you use this method to find the distance between any two points in the coordinate plane? Explain.

The Distance Formula

√(𝑥2−𝑥1)2+(𝑦2−𝑦1)

2

Prove that is longer than

a. Write coordinates: A(___,___)B(___,___)C(___,___)D(___,___)

The Distance Formula

Use the distance formula to find AB and CD.

Reflect 2a.

• When you use the distance formula, does the order in which you subtract the x-coordinates and the y-coordinates matter? Explain.

Midpoint

Essential Question: How do you use the midpoint formula to find the midpoint of a line segment in a coordinate plane?

Midpoints of Line Segments

• Use a ruler: measure the length of the line segment to the nearest millimeter.

• Find half the length of the segment.

Midpoints of Line Segments

• Record the data below.

Reflect 1a.

• Make a conjecture: If you know the coordinates of the endpoints of a line segment, how can you find the coordinates of the midpoint?

Reflect 1b.

• What are the coordinates of the midpoint of a line segment with endpoints at the origin and at the point ?

The Midpoint Formula

𝑀 ( 𝒙𝟏+𝒙𝟐

𝟐,𝒚𝟏+𝒚𝟐

𝟐 )has endpoints and Prove that the midpoint of lies in Quadrant III.

Identify : _______: _______: _______: _____

Reflect 2a.

• What must be true about and ? Show that this is the case.