Chapter 9.9, Problem 20PSC

To determine

To find:The two possible values of $\sin \angle PQS$ are to be found

Answer to Problem 20PSC

The values are $0.5$ and $0.866$

Explanation of Solution

Given information:The following information has been given

  $PQ=QR=RS=PS=15$

  $QS=15$ or $PR=15$

Midpoint of diagonals is assumed to be a point O

Formula used:The diagonals of a rhombus are perpendicular to each other

Calculation:we know that as the diagonals of a rhombus are perpendicular,

  $\sin \angle PQS=\sin \angle PQO$

Now, we know that in triangle PQO, the hypotenuse is $PQ=15$ , and the other side is $7.5$

Using Pythagoras, we can say that the third side will be

  ${7.5}^2+x^2={15}^2\to x=12.99$

Hence, using basic definitions of sine formula, we have

  $\sin \angle PQS=\frac{7.5}{15}$ or $\sin \angle PQS=\frac{12.99}{15}$

Hence, the two values of sine are $0.5$ and $0.866$