Simplification and Rectangle Trigonometry

In this exercise, we will solve two problems. Part a involves simplifying an algebraic expression, and part b is a geometric problem involving a rectangle.

Let's start with part a. We need to simplify the expression x squared minus y squared, divided by three x plus three y.

Notice that the numerator is a difference of two squares. We can factor it into x minus y times x plus y.

Now, let's look at the denominator. We can factor out a common three, giving us three times the quantity x plus y.

We can cancel the common factor of x plus y from both the top and the bottom.

This leaves us with the simplified expression x minus y divided by three.

Now let's move to part b. We have a rectangle PQRS. We are given the length of PK is fifteen centimeters, the angle P K S is thirty-seven degrees, and point K is the midpoint of the side S R.

In right-angled triangle P S K, we can find the side P S using the sine ratio, since it is opposite the thirty-seven degree angle.

Multiplying both sides by PK, which is fifteen, we get PS equals fifteen times sine of thirty-seven degrees.

Using a calculator, sine of thirty-seven degrees is approximately zero point six zero one eight. This result gives us PS as nine point zero two seven, or nine point zero three centimeters to three significant figures.