Answer:
a) The measure of each arc = 60°
b) x = 10 - 5√3
c) x = 10 - 5√3
Step-by-step explanation:
a) ∵ The Δ is equilateral
∴ m∠A = m∠B = m∠C = 60°
∵ A is the center of arc BC , B is the center of arc AC and C is the
center of the arc AB
∵ Measure any arc = measure central angle subtended by this arc
∵ m∠A = 60° , m∠B = 60° , m∠C = 60°
∴ The measure of each arc = 60°
b) ∵ ΔABC is equilateral
∵ CE ⊥ AB and bisects it
∴ AE = 10/2 = 5
∵ AC = 10 , m∠CAE = 60°
∴ sin60 = CE/CA
∴ CE = CA × sin60 ⇒ CE = 5√3
∵ CD = 10 ⇒ radius
∴ x = CD - CE = 10 - 5√3
c) Let AB and FD are two chords in the circle of center C
Where F is a point on circle C and FD passing through C
By using chord-chord theorem:
∴ FE × ED = AE × BE
∵ FE = FD - x ⇒ FD is a diameter
∴ FE = 20 - x
∴ (20 - x) × x = 5 × 5
∴ 20x - x² = 25
∴ x² - 20x + 25 = 0
∴ x = 10 - 5√3
