Question

Points C,
T,
and P
are on circle O. f m∠COT=146∘
and CT⌢=35.7
units, what is the measure of OT¯¯¯¯¯¯¯?
Round your answer to the nearest whole number.

Answer 1

To find the measure of OT¯¯¯¯¯¯¯, we can use the Law of Cosines.

Let x be the measure of OT¯¯¯¯¯¯¯. We can use the Law of Cosines formula:

(CT⌢)^2 = (CO⌢)^2 + (OT⌢)^2 - 2(CO⌢)(OT⌢)cos(m∠COT)

Substitute CT⌢ = 35.7, m∠COT = 146°, and CO⌢ = r (radius of circle O) into the equation:

35.7^2 = r^2 + x^2 - 2(r)(x)cos(146°)

1276.49 = r^2 + x^2 - 2rx(cos 146°)

Since the points are on the circle, CO⌢ = OT⌢ = x.

1276.49 = 2x^2 - 2x^2(cos 146°)

1276.49 = 2x^2(1 - cos 146°)

1276.49 = 2x^2(1 + cos 34°)

x^2 = 1276.49 / (2(1 + cos 34°))

x^2 = 288.83

x ≈ √288.83

x ≈ 17

Therefore, the measure of OT¯¯¯¯¯¯¯ is approximately 17 units (rounded to the nearest whole number).