4.2.4 Introduction to Coordinate Proof (continued) Example 3 Assigning Coordinates to a Vertex
The first figure given here is a right triangle. Since a right triangle has one right angle, position that triangle so that the vertex at the right angle is at the origin. This triangle’s legs have length m and n . So, assign one leg length m and the other length n , but it doesn’t matter which leg has which length. Here, the vertical leg is assigned length m . Therefore, the vertical leg extends to m on the y -axis, or from (0, 0) to (0, m ). And since the horizontal leg is assigned length n , this leg extends to n on the x -axis, or from (0, 0) to ( n , 0). The rectangle has four right angles, so place the rectangle so that one vertex is at the origin, one vertex is on the x -axis, and another vertex is on the y -axis.
Example 4 Writing a Coordinate Proof In this example, two methods are demonstrated for proving that the area △ DBE is one-fourth the area of △ ABC . The same conjecture is proven in each proof, but the figure’s coordinates are identified differently in each proof. In each proof, it is given that a triangle, △ ABC , contains a right angle at ∠ B . Additionally, it is given that the midpoints of two of the triangle’s sides are identified as D and E, where D is the midpoint between A and B and E is the midpoint between B and C . The first step in each proof is to position △ ABC on a coordinate plane. Since ∠ B is a right angle, position B at the origin. It follows that A must lie on the y -axis and C must lie on the x -axis. The lengths of the sides of △ ABC are unknown, so the next step is to identify variables for the lengths of AB and BC . The difference in the two proofs below is based on the lengths chosen for AB and BC .
Let AB = n and let BC = m . So, the coordinates of A must be (0, n ) and the coordinates of C must be (0, m ). Now that the coordinates of A , B , and C are identified, the coordinates of vertices D and E in △ DBE can be deduced (note that vertex B is the same point in both triangles). Since D is the midpoint of A and B , use the midpoint formula with (0, 0) and (0, n ) to find the coordinates of D . The midpoint formula can also be used to find the coordinates of E , but use (0, 0) and (0, m ) since E is the midpoint between B and C .
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