Question

In the diagram below secants PT & PU have been drawn from exterior point P such that the four arcs intercepted have the following ratio of measurements arc RS: arc ST:arc TI: arc UR= 1:4:4:3.

(a) If arc RE= X then write an equation that could be used to solve for x and find the value of x

(b) state the measure of each of the four arcs. RS, ST, TU, and UR

(c) find the measure of angle P

(d) draw segment UT what would be the measure of angle UTS?

Answer 1

• x = 30°
• RS = 30°, SR = TU = 120°, UR = 90°
• ∠P = 45°
• ∠UTS = 60°

Explanation:

(a) If RS = x, then the sum of arcs around the circle is ...

x + 4x +4x +3x = 360°

12x = 360°

x = 30°

__

(b) Based on the given ratios, the arc measures are computed from x. For example, ST = TU = 4x = 4(30°) = 120°

• RS = 30°
• ST = 120°
• TU = 120°
• UR = 90°

__

(c) Angle P is half the difference of arcs TU and RS:

∠P = (TU -RS)/2 = (120° -30°)/2

∠P = 45°

__

(d) Inscribed angle UTS is half the measure of the arc it intercepts. Arc RU has the measure (30° +90°) = 120°, so the measure of UTS is ...

∠UTS = 120°/2 = 60°