Question

Lesson Performance Task


The figure shows the basic design of a Wankel rotary engine. The triangle is equilateral, with sides measuring 10 inches. An arc on each side of the triangle has as its center the vertex on the opposite side of the triangle.
In the figure, the arc ADB is an arc of a circle with its center at C.




a. Use the sketch of the engine. What is the measure of each arc

along the side of the triangle?


b. Use the relationships in an equilateral triangle to find the value

of x. Explain.


c. Use the Chord-Chord Product Theorem to find the value of x.

Explain.





I know that a. is 180 but I don't understand b or c

My teacher told me that I need to draw the circle because of " In the figure, the arc ADB is an arc of a circle with its center at C." and that confused me. The pic with the red circle shows kinda what he told me to do.

Answer 1

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