The FRT triangle is defined by the coordinates of its vertices: F (2; -2), R (2; 3)

The FRT triangle is defined by the coordinates of its vertices: F (2; -2), R (2; 3), T (-2; 1). a) Prove that triangle FRT is isosceles. b) Find the height drawn from vertex F.

F (2; -2), R (2; 3), T (-2; 1)

a) Let us find the lengths of the sides of the triangle by the formula: if points A and B are given by their coordinates A (x; y) and B (x1; y1), then | AB | = √ [(x1 – x) ^ 2 + y1 – y) ^ 2].

| FR | = √ [(2 – 2) ^ 2 + (3 – (-2)) ^ 2] = √ [0 + 25] = 5 units.

| RT | = √ [((- 2) – 2) ^ 2 + (1 – 3) ^ 2] = √ [16 + 4] = √20 = 2√5 units.

| FT | = √ [((- 2) – 2) ^ 2 + (1 – (-2) ^ 2] = √ [16 + 9] = √25 = 5 units.

Note that | FR | = | FT | = 5 units, so in a given triangle, two sides are equal in length, which is the definition of an isosceles triangle. Ch.t.d.

b) So, we have proved that our triangle is isosceles, so the height FO drawn to the side TR is also its median (by the property of an isosceles triangle), and therefore point O divides the side TR in half, i.e. TO = TR.

Knowing the coordinates of the points T (-2; 1) and R (2; 3), we can find the coordinate of the point O, and then the length of FO.

The coordinates of the point O, as the midpoint of the segment TR: (-2 + 2/2; 1 + 3/2) = O (0; 2).

We have F (2; -2) and O (0; 2).

| FО | = √ [(0 – 2) 2 + (2 – (-2)) 2] = √ [4 + 16] = √20 = 2√5 units.