) RSTU is a cyclic quadrilateral in a circle of centre O.If RUT is 40°, what are the two values of <ROT (convex and concave)?

(b) A RHC has a triangular catchment’s area PQR such that PQ = 22.1Km; PR = 13.5Km and <RPQ =56°. Find <PRQ.

In order to find the values of <ROT, we need to use the properties of cyclic quadrilaterals. In a cyclic quadrilateral, the opposite angles formed by the sides of the quadrilateral with the circle's center are supplementary, which means their sum is 180 degrees.

Here, we are given that RUT is 40 degrees. Since RUT is an angle with the center of the circle, the opposite angle to RUT will be <ROT. Therefore, <ROT + 40 = 180.

To solve for <ROT, we can subtract 40 from 180: <ROT = 180 - 40 = 140 degrees.

So, the convex value of <ROT is 140 degrees.

The concave value of <ROT can be calculated by subtracting the convex angle from 360 degrees. Therefore, <ROT (concave) = 360 - 140 = 220 degrees.

Now, let's move on to the second question about the triangular catchment area PQR.

Here, we are given that PQ = 22.1 km, PR = 13.5 km, and <RPQ = 56 degrees. We want to find <PRQ.

To find <PRQ, we can use the angle sum property of triangles, which states that the sum of the angles in a triangle is always 180 degrees.

In triangle PQR:
<PQR + <RPQ + <PRQ = 180.

We are given that <RPQ = 56 degrees. Let's substitute the given values into the equation:
56 + <PQR + <PRQ = 180.

To find <PRQ, we can rearrange the equation:
<PQR + <PRQ = 180 - 56,
<PQR + <PRQ = 124.

Since we know that the sum of the angles in a triangle is 180, we can subtract the given angle <PQR from 124:
124 - <PQR = <PRQ.

Therefore, <PRQ = 124 - <PQR.

Now you can substitute the given value of <PQR into the equation to find the measure of <PRQ.