Question

Francesca is writing a coordinate proof to show that the diagonals of a rectangle divide the rectangle into four triangles of equal area.

A rectangle graphed on a coordinate plane. The vertices of rectangle are labeled as P Q R and S. The vertex labeled as P lies on begin ordered pair 0 comma 0 end ordered pair. The vertex labeled as Q lies on begin ordered pair 0 comma 2 y end ordered pair. The coordinate of vertex R is left blank. The vertex labeled as S lies on begin ordered pair 2 x comma 0 end ordered pair. Two diagonals P R and Q S intersect each other at a point labeled as T.


Enter your answers in the boxes to complete Francesca's proof.

Since PQRS is a rectangle, the coordinates of R are (
, 2y).

Since diagonals of a rectangle bisect each other, the coordinates of T are (x,
).

In △PTS, the length of the base is 2x and the height is y. So an expression for the area of △PTS is ​xy​ .

In △PTQ, the length of the base is 2y and the height is x. So an expression for the area of △PTQ is
.

In △RTQ, the length of the base is
and the height is y. So an expression for the area of △RTQ is xy.

To find the area of △RTS, the length of the base is 2y and the height is
. So an expression for the area of △RTS is ​xy​ .

Comparing the four areas shows that the diagonals of a rectangle divide the rectangle into four triangles of equal areas.

Answer 1

Since PQRS is a rectangle, the coordinates of R are (2x, 2y).

Since diagonals of a rectangle bisect each other, the coordinates of T are (x, y).

In △PTQ ... area = xy.

In △RTQ, the length of the base is 2x and the height is y. So area = xy.

In △RTS ... the height is x. So area = xy.