4.2.5 Isosceles and Equilateral Triangles (continued)
In this example it is given that the three sides of △ PQR are congruent. Therefore, △ PQR is an equilateral triangle. So, by the Equilateral Triangle Theorem, it can be assumed that △ PQR is also equiangular. It follows that the measure of each angle in △ PQR must be equal to 60°, since △ PQR is equiangular. So, set the expression given for m ∠ P , (2 x + 10)°, equal to 60° and solve the equation for x . The triangle given here has three congruent angles. Therefore, the triangle is equiangular. So, by the Equiangular Triangle Theorem, it can be assumed that the triangle is also equilateral. Equilateral triangle must have three congruent sides, by definition. So, the expressions given for the lengths of two of the triangle’s sides can be set equal to each other. 2 t + 13 = 5 t + 1 Solve this equation for t . Remember, when writing a coordinate proof, the given figure can be placed anywhere on a coordinate plane. Since the figure given here is a right triangle, place the right angle at the origin where one side of the triangle is placed on the x -axis and another side is placed on the y -axis. And since it is given that AB and BC are congruent, the right angle must be ∠ B . So, draw △ ABC so that ∠ B is at the origin, A is on the y -axis, and C is on the x -axis. Here, the length of AC is chosen to be 2 s and the length of BC is chosen to be 2 r . So, the coordinates of A and C are (0, 2 s ) and (2 r , 0), respectively. D is given to be the midpoint of AC . So, use the midpoint formula to find the coordinates of D . To prove that △ BDC is isosceles, show that the lengths of two of the sides are equal. Use the distance formula to find the lengths of BD and CD .
Example 4 Using Coordinate Proof
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