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1-1b: The Coordinate Plane - Distance Formula & Pythagorean Theorem CCSS GSE: M(G&M)–10–9 Solves problems on and off the coordinate plane involving distance, midpoint, perpendicular and parallel lines, or slope M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or across disciplines or contexts (e.g., Pythagorean Theorem Example: Find the measure of AB. A Point A is at 1.5 and B is at 5. So, AB = 5 - 1.5 = 3.5 B Example • Find the measure of PR • Ans: |3-(-4)|=|3+4|=7 • Would it matter if I asked for the distance from R to P ? Ways to find the length of a segment on the coordinate plane • 1) Pythagorean Theorem- Can be used on and off the coordinate plane •2) Distance Formula – only used on the coordinate plane 1) Pythagorean Theorem* * Only can be used with Right Triangles What are the parts to a RIGHT Triangle? Hypotenuse- Side across from the 1. Right angle right angle. Always the longest 2. 2 legs side of a right triangle. 3. Hypotenuse LEG Right angle Leg – Sides attached to the Right angle Pythagorean Formula (leg) (leg) (hypotenuse) 2 2 2 Example of Pyth. Th. on the Coordinate Plane Make a right Triangle out of the segment (either way) Find the length of each leg of the right Triangle. Then use the Pythagorean Theorem to find the Original segment JT (the hypotenuse). Find the length of CD using the Pythagorean Theorem We got 10 by | 6 - - 4| 82 102 DC 2 10 64 100 DC 2 164 DC 2 164 DC 12.8 8 We got 8 by | -4 – 4| Ex. Pythagorean Theorem off the Coordinate Plane • Find the missing segment- Identify the Leg parts of the triangle 5 in Leg 2 + Leg 2 = Hyp 2 Ans: 5 2 + X 2 = 13 2 13 in 25 + X 2 = 169 hyp 2 X = 144 X = 12 in Leg 2) Distance Formula Lets Use the Pythagorean Theorem d= x2 x1 y2 y1 2 2 J (-3,5) T (4,2) x1, y1 x2, y2 Identify one as the 1st point and one as the 2nd. Use the corresponding x and y values (4-(-3))2 + (2-(5))2 (4+3)2 + (2-5)2 (7)2 +(-3)2 49+9 = 58 ~ 7.6 ~ Example of the Distance Formula • Find the length of the green segment Ans: 109 or approximately 10.44 ( ) Congruent Segments • Segments that have the same length. If AB & XY have the same length, Then AB=XY, but AB XY Symbol for congruent Assignment