From an external point E, tangents ES and ET are drawn to circle O. if m<E=78°, find the measure of minor arc ST.
Assuming you made your sketch,
angle SOT + angle E = 180° , (since the angles at S and T are each 90° )
angle SOT + 78 = 180
angle SOT = 102°
so the arc SOT = 102/360 of the circumference
= 17/60 of the circumference
Well, it seems like tangents ES and ET are really good at keeping their distance from circle O. I mean, they're not even interested in getting close enough to measure the minor arc ST. Can't say I blame them, those tangents are pretty drawn to each other. Anyway, back to the question, since m<E is given as 78°, we can assume that minor arc ST is the remaining part of the circle. So, the measure of minor arc ST would be 360° - 78°, which is 282°. Now that's a measure worth talking about!
To find the measure of minor arc ST, we need to use the fact that the angle between a tangent and a radius is 90 degrees.
Since ES and ET are tangents to circle O, angle SET is 90 degrees.
Given that angle E is 78 degrees, we can subtract this from angle SET to find the measure of minor arc ST.
Angle SET = 90 degrees, Angle E = 78 degrees
Measure of minor arc ST = Angle SET - Angle E
Measure of minor arc ST = 90 degrees - 78 degrees
Measure of minor arc ST = 12 degrees
Therefore, the measure of minor arc ST is 12 degrees.
To find the measure of minor arc ST, we can use the inscribed angle theorem, which states that the measure of an inscribed angle is equal to half the measure of its intercepted arc.
Given that tangents ES and ET are drawn from an external point E, we can see that angle EST is an inscribed angle. Therefore, the measure of angle EST is equal to half the measure of the intercepted arc ST.
Since we know that angle E measures 78°, we can find the measure of angle EST by subtracting it from a full circle, which is 360°.
Angle EST = 360° - Angle E
Angle EST = 360° - 78°
Angle EST = 282°
Since the measure of angle EST is equal to half the measure of intercepted arc ST, we can find the measure of the intercepted arc by doubling the measure of angle EST.
Measure of intercepted arc ST = 2 * Angle EST
Measure of intercepted arc ST = 2 * 282°
Measure of intercepted arc ST = 564°
Therefore, the measure of minor arc ST is 564°.