Question

До кола проведено дотичну AB (B-точка дотику) та січна AC, що проходить через центр О кола. Знайдіть градусну міру кута COB , якщо ∠OAB=35°

Answer 1

∠АВО = 90° т.к. касательная АВ перпендикулярна радиусу ОВ.
∠СОВ - внешний угол для треугольника АВО при вершине О, он равен сумме двух внутренних углов, не смежных с ним, поэтому
∠СОВ = ∠АВО + ∠ОАВ = 90° + 35° = 125°
Ответ: 125°

Answer 2

Given Information:

- The circle has a point of tangency at point B on the circle. - The line AC is a secant that passes through the center O of the circle. - The measure of angle OAB is 35°.

To find:

- The measure of angle COB.

Solution:

To find the measure of angle COB, we need to use some properties of tangents and secants.

Property 1: The measure of an angle formed by a tangent and a chord drawn from the point of tangency is equal to half the measure of the intercepted arc.

In this case, angle COB is formed by the tangent AB and the chord AC. The intercepted arc is the arc AC. So, we can say that angle COB is equal to half the measure of arc AC.

Property 2: The measure of an angle formed by two intersecting chords is equal to half the sum of the measures of the intercepted arcs.

In this case, angle COB is also formed by the secant AC and the chord AB. The intercepted arcs are arc AC and arc AB. So, we can say that angle COB is equal to half the sum of the measures of arcs AC and AB.

Now, let's use these properties to find the measure of angle COB.

First, we need to find the measures of arcs AC and AB.

Since angle OAB is given as 35°, we know that angle OAC (the inscribed angle that intercepts arc AC) is also 35°. And since angle OAB is formed by the tangent AB and the chord AC, we can say that arc AC has a measure of 2 * angle OAC = 2 * 35° = 70°.

Similarly, since angle OAB is formed by the secant AC and the chord AB, we can say that arc AB has a measure of 2 * angle OAB = 2 * 35° = 70°.

Now, let's find the measure of angle COB using the properties mentioned above.

According to Property 1, angle COB is equal to half the measure of arc AC. So, angle COB = 0.5 * 70° = 35°.

According to Property 2, angle COB is also equal to half the sum of the measures of arcs AC and AB. So, angle COB = 0.5 * (arc AC + arc AB) = 0.5 * (70° + 70°) = 0.5 * 140° = 70°.

Therefore, the measure of angle COB is 35° or 70°, depending on which property you use.

In conclusion, the measure of angle COB is either 35° or 70°, depending on the interpretation of the properties used.