4.7 Triangles and Coordinate Proof. Objectives:. Place geometric figures in a coordinate plane. Write a coordinate proof. Assignment: 2-26 even. Placing Figures in a Coordinate Plane.
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One vertex is at the origin, and three of the vertices have at least one coordinate that is 0.Ex. 1: Placing a Rectangle in a Coordinate Plane
One side is centered at the origin, and the x-coordinates are opposites.Ex. 1: Placing a Rectangle in a Coordinate Plane
A right triangle has legs of 5 units and 12 units. Place the triangle in a coordinate plane. Label the coordinates of the vertices and find the length of the hypotenuse.Ex. 2: Using the Distance Formula
One possible placement is shown. Notice that one leg is vertical and the other leg is horizontal, which assures that the legs meet as right angles. Points on the same vertical segment have the same x-coordinate, and points on the same horizontal segment have the same y-coordinate.Ex. 2: Using the Distance Formula
d = √(x2 – x1)2 + (y2 – y1)2
= √(12-0)2 + (5-0)2
= 13Ex. 2: Using the Distance Formula
In the diagram, hypotenuse. ∆MLO ≅ ∆KLO). Find the coordinates of point L.
Solution: Because the triangles are congruent, it follows that ML ≅ KL. So, point L must be the midpoint of MK. This means you can use the Midpoint Formula to find the coordinates of point L.Ex. 3 Using the Midpoint Formula
L (x, y) = x hypotenuse. 1 + x2, y1 +y2
=160+0 , 0+160
= (80, 80)
Simplify.Ex. 3 Using the Midpoint Formula
Given: Coordinates of vertices of hypotenuse. ∆POS and ∆ROS.
Prove SO bisects PSR.Ex. 4: Writing a Plan for a Coordinate Proof
Right hypotenuse. ∆QBC has leg lengths of h units and k units. You can find the coordinates of points B and C by considering how the triangle is placed in a coordinate plane.
Point B is h units horizontally from the origin (0, 0), so its coordinates are (h, 0). Point C is h units horizontally from the origin and k units vertically from the origin, so its coordinates are (h, k). You can use the Distance Formula to find the length of the hypotenuse QC.Ex. 5: Using Variables as Coordinates
C (h, k)
Q (0, 0)
B (h, 0)
OC = hypotenuse. √(x2 – x1)2 + (y2 – y1)2
= √(h-0)2 + (k - 0)2
= √h2 + k2Ex. 5: Using Variables as Coordinates
C (h, k)
Q (0, 0)
B (h, 0)
Given: Coordinates of figure OTUV hypotenuse.
Prove ∆OUT ∆UVO
Coordinate proof: Segments OV and UT have the same length.
OV = √(h-0)2 + (0 - 0)2=h
UT = √(m+h-m)2 + (k - k)2=hEx. 5 Writing a Coordinate Proof
Horizontal segments UT and OV each have a slope of 0, which implies they are parallel. Segment OU intersects UT and OV to form congruent alternate interior angles TUO and VOU. Because OU OU, you can apply the SAS Congruence Postulate to conclude that ∆OUT ∆UVO.Ex. 5 Writing a Coordinate Proof