In the diagram, PQRS is a circle. = . ?SPR = 26° and the interior angles of PQS are in the ratio 2:3 :3. Calculate: (i) PQR; (ii) RPQ; (iii) PRQ (b) The coo...

In the diagram, PQRS is a circle.  $$  =  $$ . ?SPR = 26° and the interior angles of PQS are in the ratio 2:3 :3.

Calculate: (i) PQR; (ii) RPQ; (iii) PRQ

(b) The coordinates of two points P and Q in a plane are (7, 3) and (5, x) respectively, where X is a real number.

If |PQ| = 29 units, find the value of x.

Answer Details

(a)

(i) Since PQRS is a circle and |PQ| = |QS|, we have ?PQR = ?QRS (angles subtended by equal chords are equal).

Let ?PQS = 2x. Then, ?PQR = 3x and ?QRS = 3x.

We know that the sum of interior angles of a triangle is 180°. So, in ?PQS, we have:

2x + 3x + 3x = 180°

8x = 180°

x = 22.5°

Now, in ?PQR, we have:

?PQR + ?QPR + ?RPQ = 180° (sum of interior angles of a triangle)

3x + 90° + 26° = 180° (since ?QPR is a right angle)

3x = 64°

x = 21.33°

Therefore, PQR = 3x = 64°.

(ii) RPQ = 180° - ?PQR - ?QPR = 180° - 64° - 90° = 26°.

(iii) PRQ = 180° - ?PQR - ?RPQ = 180° - 64° - 26° = 90°.

(b) Using the distance formula, we can find the distance between P and Q:

|PQ|² = (5-7)² + (x-3)²

|PQ|² = 4 + (x-3)²

Since |PQ| = 29 units, we have:

29² = 4 + (x-3)²

841 = (x-3)² + 4

837 = (x-3)²

Taking the square root of both sides, we get:

x-3 = ±?837

x = 3 ± ?837

Since x is a real number, we take the positive square root:

x = 3 + ?837

Therefore, the value of x is 3 + ?837 units.