) A chord AB of length 13cm is drawn in a circle centre O at a distance of 5cm from O. Find the radius of the circle. Stop at 1 d.p.(b) TA and TB are tangents to a circle centre O. TOC is a straight line and <ATB = 56°. Calculate <ACO.
11.9cm
To find the radius of the circle and the measure of angle ACO, we can use some geometric principles and trigonometry.
(a) Finding the radius:
1. In a circle, the radius is always perpendicular to the chord that passes through its center. Therefore, the line joining the center of the circle (O) to the midpoint of the chord (AB) is perpendicular to AB.
2. Since AB is a chord of length 13 cm and is drawn at a distance of 5 cm from the center O, the line segment joining O to the midpoint of AB splits AB into two segments of equal length.
3. Thus, the length of each segment is (13/2) cm.
4. By using the Pythagorean theorem, we can find the distance from the center O to the midpoint of AB. Let's call this distance "x".
- Using (13/2)^2 + x^2 = 5^2
- Solving this equation gives x ≈ 4.87 cm.
5. Therefore, the radius of the circle is approximately 4.9 cm (rounded to 1 decimal place).
(b) Finding the measure of angle ACO:
1. Angle ACO is an inscribed angle, and an inscribed angle is equal to half the measure of its intercepted arc.
2. Angle ATB is given as 56°, and since TA and TB are tangents to the circle, they intersect at the point of tangency, forming a right angle. Hence, angle ATB is a right angle.
3. The intercepted arc of angle ATB is the arc subtended by the chord AB.
4. The measure of the intercepted arc is equal to double the measure of angle ATB because an inscribed angle is equal to half the intercepted arc.
5. Therefore, the measure of the intercepted arc is 2 * 56° = 112°.
6. Angle ACO is half the measure of the intercepted arc, so angle ACO = 112° / 2 = 56°.
Therefore, the measure of angle ACO is 56°.