Andrew is writing a coordinate proof to show that the triangle formed by connecting the midpoints of the sides of an isosceles triangle is itself an isosceles triangle. He starts by assigning coordinates as given. An isosceles triangle is graphed on a coordinate plane. The horizontal x-axis and y-axis are solid, and the grid is hidden. The vertices are labeled as E, D, and F. The vertex labeled as E lies on begin ordered pair 0 comma 0 end ordered pair. The vertex labeled as D lies on begin ordered pair 2 a comma 2 b end ordered pair. The vertex labeled as F lies on begin ordered pair 4 a comma 0 end ordered pair. Midsegment is drawn between DE and DF labeled as PQ. Midsegment is drawn between FE and DF labeled as QR. Midsegment is drawn between DE and EF labeled as PR. The midsegments form triangle P Q R. Enter the answers in the boxes to complete the coordinate proof. P is the midpoint of DE¯¯¯¯¯. Therefore, the coordinates of P are ( , b). Q is the midpoint of DF¯¯¯¯¯. Therefore, the coordinates of Q are (3a, ). R is the midpoint of EF¯¯¯¯¯. Therefore, the coordinates of R are ( , ). The length of PR¯¯¯¯¯ is a2+b2−−−−−−√ . The length of QR¯¯¯¯¯ is a2+b2−−−−−−√. Comparing the expressions for the lengths of PR¯¯¯¯¯ and QR¯¯¯¯¯ shows that the lengths are equal. Therefore, △PQR is isosceles.
