Question

Given mCT=26 and m/CAT =124° find the length of CA, the radius in Circle A. Use

π = 3.14 in your calculation and round to the nearest tenth.

Answer 1

`\text{length of \textit{CA}} \ = \ 12.0 \ \ \ (\text{nearest tenth})`

Explanation:

Radian measure is the ratio of the length of a circular arc to its radius.

A radian is the measurement of the central angle which subtends an arc whose length is equal to the length of the radius of the circle.

In the case of the unit circle, as shown in the figure below, one radian is the angle of the sector with a radius of 1 and circular arclength of 1.

Following this definition, the magnitude, in radians, of one complete revolution of a unit circle is the circumference of the unit circle divided by its radius,
`\displaystyle(2\pi)/(1)` or
`2\pi`. Thus,
`2\pi` radians is equal to
`360^(\circ)` degrees. Alternatively, one radian is equal to
`\displaystyle(180^(\circ))/(\pi) \ \approx \ 57.296^(\circ) \ \ \ (3 \ d.p.)`.

Since radian measure is defined as the ratio of the arc length of a sector to its radius, hence

`\displaystyle(s)/(r) \ = \ \theta \\\\ s \ = \ r\theta`

where
`s` is the arclength,
`r` is the radius, and
`\theta` is the central angle, in radians.

Therefore, the length of CA is

`\displaystyle(s)/(\theta) \ = \ r \\ \\ r \ = \ \displaystyle\frac{26}{124^(\circ) \ * \ \displaystyle(\pi)/(180^(\circ)) \ \text{rad}} \\ \\ r = \displaystyle(26 \ * \ 45)/(31 \ * \ 3.14) \\ \\ r\ = \ 12.0 \ \ \ \ \left(\text{nearest tenth}\right)`