Answer:
The measure of the central angle is 35° to the nearest degree
The area of the triangle formed is 28.7 inches²
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