The diameter of circle C is 18 cm. mAD is 60°. Answer the following:
a) Find m<DAB.Explain.
b) Find <mDBA. Explain.
c) Find m<BDA. Explain.
d) Find the length L of AD.
To find the values in the given situation, we need to apply the properties of circles and angles.
a) To find m<DAB:
- We know that circle C has a diameter of 18 cm.
- The diameter of a circle passes through the center of the circle, splitting it into two equal halves.
- The angle subtended by the diameter at any point on the circumference is a right angle (90°).
- Since mAD is given as 60°, we can conclude that <mDAB is half of mAD. Therefore, <mDAB is equal to 30°.
b) To find <mDBA:
- We know that in a circle, angles subtended by the same arc are equal.
- <mDAB and <mDBA both subtend the same arc, which is the arc DA on the circumference.
- Therefore, <mDAB and <mDBA are equal. Since we found <mDAB to be 30° in part (a), <mDBA is also 30°.
c) To find m<BDA:
- We know that the angles of a triangle add up to 180°.
- Triangle ABD is a triangle formed by the points A, B, and D on the circumference of the circle.
- We already know that <mDAB is 30° from part (a), and <mDBA is 30° from part (b).
- So, m<BDA is equal to 180° - (30° + 30°) = 120°.
d) To find the length L of AD:
- Since the diameter of circle C is 18 cm, AD is a chord of the circle.
- The relationship between the length of a chord (AD) and the diameter (C) is given by the theorem: L = 2r sin(<mDAB/2).
- We already know that the diameter is 18 cm, and we found <mDAB to be 30° in part (a).
- Plugging these values into the formula: L = 2(9)sin(30°/2) = 2(9)sin(15°).
To find the exact value of sin(15°), we can use the half-angle formula for sine: sin(15°) = sqrt[(1 - cos(30°))/2].
Using trigonometric tables or a calculator, we can find the value of sin(15°) to be approximately 0.2588.
Substituting this value into the formula, we get: L = 2(9)(0.2588) = 4.6584 cm (rounded to four decimal places).
Therefore, the length of AD, which is represented by L, is approximately 4.6584 cm.