Distance.

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Presentation transcript:

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) Plot the points A and B in the coordinate plane on the right. Draw 𝐴𝐵 Draw a vertical line through point A Draw a horizontal line through point B 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: 𝑎 2 + 𝑏 2 = 𝑐 2

Distance in the Coordinate Plane A(2,5) B(-4,-3) Pythagorean Theorem: 8 2 + 6 2 = 𝐴𝐵 2 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 ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 Prove that 𝐶𝐷 is longer than 𝐴𝐵 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 (𝑎,𝑏)?

The Midpoint Formula 𝑃𝑄 has endpoints 𝑃 −4,1 and 𝑄(2,−3) 𝑀 𝒙 𝟏 + 𝒙 𝟐 𝟐 , 𝒚 𝟏 + 𝒚 𝟐 𝟐 𝑃𝑄 has endpoints 𝑃 −4,1 and 𝑄(2,−3) Prove that the midpoint 𝑀 of 𝑃𝑄 lies in Quadrant III. Identify 𝑥 1 , 𝑥 2 , 𝑦 1 , 𝑎𝑛𝑑 𝑦 2 𝑥 1 : _______ 𝑥 2 : _______ 𝑦 1 : _______ 𝑦 2 : _____

Reflect 2a. What must be true about 𝑃𝑀 and 𝑄𝑀? Show that this is the case.