Acircle with radius 10 in has radii drawn to the endpoints of a 6 in chord. what is the measure of the central angle, to the nearest degree? what is the area of the triangle formed, to the nearest tenth. show work

The measure of the central angle is 35° to the nearest degree

The area of the triangle formed is 28.7 inches²

Step-by-step explanation:

* Lets revise the cosine rule to solve the question

- In Δ ABC:

# AB² = BC² + AC² - 2(BC)(AC) cos∠C

- If you want to find m∠C, we can rearrange the rule

# cos∠C = BC² + AC² - AB²/2(BC)(AC)

* Now lets solve the problem

- The length of the radius of the circle is 10 inches

- The length of the chord is 6 inches

- The chord and the two radii drawn to the endpoints of the chord

 formed an isosceles triangle and the angle between the two radii

 is the central angle

- Lets use the cosine rule to find the measure of the central angle

- Let the name of the central angle is Ф

∵ cos Ф = r² + r² - (chord)²/2(r)(r)

∵ r = 10 inches and the chord = 6 inches

∴ cos Ф = (10)² + (10)² - (6)²/2(10)(10)

∴ cos Ф = (100+ 100 - 36)/200

∴ cos Ф = 164/200 = 41/50

- Find the measure of the Ф by using cos^-1 Ф

∴ m∠Ф = cos^-1 (41/50) ≅ 35°

* The measure of the central angle is 35° to the nearest degree

- To find the area of the triangle we will use the sine rule

# Area of any triangle by using the sine rule is

  A = 1/2 × (s1 × s2) × sin the included angle between them

∵ s1 = r , s2 = r and the angle between them is the central angle Ф

∴ Area of the triangle = 1/2 (r × r) × sin Ф

∵ r = 10 and Ф = 35°

∴ Area of the triangle = 1/2 × (10 × 10) × sin 35°

∴ Area of the triangle = 50(sin 35°) ≅ 28.7 inches²

* The area of the triangle formed is 28.7 inches²