Chapter 5.4, Problem 30WE

To determine

To prove: $△QSK$ is isosceles.

Explanation of Solution

Given:

Rectangle QRST and parallelogram $▱RKST$

Concept used:

Diagonals of rectangles are equal and bisect each other.

The opposite sides of a parallelogram are equal.

Proof:

Construct a rectangle QRST and parallelogram $▱RKST$ .

  

Now join QS to form triangle $△QSK$.

  

It is to prove that $△QSK$ is isosceles which can be proved by showing any two sides of the triangle are of equal length.

In rectangle QRST , TR and SQ are the diagonals of rectangle. As the diagonals of the rectangle are of equal length we can say that $SQ=RT$ .

Now in the parallelogram RKST , RT and SK are parallel sides of the parallelogram, and thus $RT=KS$ .

Therefore, $SQ=KS$

Thus, in $△QSK$ , two sides are equal ( $SQ=KS$ ), so we can say that $△QSK$ is an isosceles triangle.