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CC H Aim #4: How do we prove a right using coordinate geometry and what is the criterion for perpendicularity? Do Now: 1. In ΔABC, AC = 4, AB = 7, and BC = 5. Is ΔABC a ? If so, name the right .

2. In ΔDEF, DE = 5, DF = , and EF = 5. Is ΔDEF a right triangle? If so, name the .

1. a) Plot points O(0,0), A(6,4) and B(-2,3) on the coordinate and construct ΔAOB. b) Find the of the three sides of ΔAOB and determine whether it is a right Δ by applying the Pythagorean .

c) Determine the slopes of segments OA and OB.

d) Find the sum of the product of the x-values and the product of the y-values of the two coordinates A and B. 2. a) Plot points O(0,0), P(3,-1) and Q(2,3) on the coordinate plane and construct ΔOPQ. b) Find the lengths of the three sides of ΔOPQ and determine whether it is a right Δ by applying the Pythagorean theorem.

c) Determine the slopes of segments OP and OQ.

d) Find the sum of the product of the x-values and the product of the y-values of the two coordinates P and Q.

3. a) Plot points O(0,0), R(4,6) and S(-3,-2) on the coordinate plane. b) Determine whether OR and OS are using slope. Explain why or why not.

c) Find the sum of the product of the x-values and the product of the y-values of the two coordinates R and S. 4. a) Plot points O(0,0), T(-6, 3) and U(2, 4) on the coordinate plane. b) Determine whether OT and OU are perpendicular using slope.

c) Find the sum of the product of the x-values and the product of the y-values of the two coordinates T and U.

Criterion for Perpendicularity:

Given O(0,0), A(a1,a2) and B(b1,b2). To determine if OA Τ OB, ______

5. ΔABC has vertices A(x, 1), B(-3,5), and C(1,4). a) Determine and state a value for x that would make ΔABC a right triangle.

*b) Prove that ΔABC is a right triangle.

*Two methods to prove a right triangle: 1) Show two sides of the triangle are perpendicular demonstrating slopes are opposite reciprocals. 2) Calculate of all three sides and test the Pythagorean Theorem. 6. Given points O(0,0), A(3,1) and B(-2,6), prove OA is perpendicular to OB: a) Using their slopes.

b) Using the perpendicularity criterion.

c) Given points P(1,-1), Q(-4,4) and R(-2,-2), prove PR is perpendicular to QR without using the pythagorean theorem, perpendicularity criterion, or finding slope.

Lets Sum it Up

If O(0,0), A(a1,a2) and B(b1,b2), OA Τ OB if a1,b1 + a2,b2 = 0.

If lines l1 and l2 are perpendicular then their slopes are negative reciprocals of each other. Name______CC Geometry H Date ______HW #4 1. Prove using coordinate geometry that ΔABC is a right triangle given A(-2,-2), B(5,-2) and C(-2,22).

2. Use the perpendicularity criteria of segments through the origin to determine if OA and OB are perpendicular.

a. A(-3,-4), B(4,3)

b. A(8,9), B(18,-16)

3. Given points O(0,0), S(2,7) and T(7,-2), where OS is perpendicular to OT, will the images of the segments be perpendicular if the three points O, S, and T are translated four units to the right and eight units up? Explain your answer.

4. In #1 from class, we saw that OA was perpendicular to OB for O(0,0), A(6,4) and B(-2,3). Suppose P(5,5), Q(11,9) and R(3,8). Are segments PQ and PR perpendicular? Explain without using or the Pythagorean theorem. Review: 5. A robot that picks up tennis balls is on a straight path from (8,6) towards a ball at (-10,-5) . The robot picks up a ball at (-10,-5), then turns 90ᵒ right. What are the coordinates of a that the robot can move towards to pick up the last ball?

6. Gerry thinks that the points (4,2) and (-1,4) form a perpendicular to a line with slope 4. Do you agree? Why or why not?

7. In ΔBCD below, BA is drawn from B to point A on DC such that BC ≅ BA.

In ΔDAB, m≮D = x, m≮DAB = 5x – 30, m≮DBA = 3x – 60, AB = 6y – 8 and BC = 4y – 2. a) Find m≮D b) Find m≮BAC

c) Find the of BC d) Find the length of DC