Question 1206533: AB is a chord of a circle and the tangents A and B meet at T. C is a point on the minor arc AB. If ∠ATB = 54° and ∠CBT = 23°, calculate ∠CAT.
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Angle ATB is 54 degrees.
Since AT and BT intersect the circle, the measure of angle ATB is half the difference between the measures of major arc AB and minor arc AB. The sum of the measures of those major and minor arcs is 360 degrees. Simple calculations show minor arc AB is 126 degrees and major arc AB is 234 degrees.
Angle CBT is 23 degrees. Since its vertex is on the circle, its measure is half the measure of the arc it cuts off. So the measure of arc CB is 46 degrees.
The measure of arc AB is 126 degrees, and the measure of arc CB is 46 degrees; so the measure of arc AC is 80 degrees.
Angle CAT has its vertex on the circle, so its measure is half the measure of the arc AC that it cuts off. That arc measure is 80 degrees, so the angle measure is 40 degrees.
ANSWER: 40 degrees
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And here is an alternate path to the final answer, once we have determined that the measure of arc AB is 126 degrees.
Angles CAT and CBT together cut off minor arc AB. Since the measure of arc AB is 126 degrees, the sum of the measures of angles CAT and CBT is 63 degrees. Then, since the measure of angle CBT is 23 degrees, the measure of angle CAT is 40 degrees.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Answer: 40 degrees
Explanation
This is likely what the diagram looks like.
The diagram was made with GeoGebra
Let D be the center of the circle. Draw segments DA and DB which are radii.
Focus only on quadrilateral DATB.
For any quadrilateral, the four inside angles must always add to 360 degrees.
D+A+T+B = 360
D+90+54+90 = 360
D+234 = 360
D = 360-234
D = 126
This is the measure of angle ADB. It is also the measure of minor arc AB. Central angles subtending an arc will have the same measure.
Note that angles A and B are 90 degrees each since they are at points of tangency.
Let E be some point on the circle that's not on minor arc AB. Refer to the diagram below.
Major arc AEB and minor arc AB glue together to form the entire circle. There are no gaps and no overlaps between the arcs. The arcs partition the circle's circumference.
Because of this we know that:
(major arc AEB) + (minor arc AB) = 360
(major arc AEB) + 126 = 360
major arc AEB = 360 - 126
major arc AEB = 234
Now we'll use the inscribed angle theorem to determine inscribed angle ACB.
angle ACB = (1/2)*(major arc AEB)
angle ACB = (1/2)*(234)
angle ACB = 117
We'll use this later after a slight detour in the next section.
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Draw a segment connecting points D and T.
Quadrilateral DATB is split into triangle DAT and triangle DBT.
They are both right triangles due to the points of tangency.
We can use the hypotenuse leg (HL) theorem to prove those right triangles are congruent, and it consequently means AT = TB.
Since AT = TB, we determine that triangle BAT is isosceles.
Furthermore, it means angle ABT = angle BAT since they are the base angles. The congruent base angles are opposite the congruent sides.
Focus only on triangle ATB.
The three inside angles must add to 180 degrees.
A+T+B = 180
x+54+x = 180
2x+54 = 180
2x = 180-54
2x = 126
x = 126/2
x = 63
Base angles ABT and BAT are 63 degrees each.
Then,
(angle CBA) + (angle CBT) = angle ABT
angle CBA = (angle ABT) - (angle CBT)
angle CBA = (63) - (23)
angle CBA = 40
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The conclusions of the first two sections are
angle ACB = 117
angle CBA = 40
These represent angles C and B in triangle ABC.
Focus only on triangle ABC.
A+B+C = 180
A+117+40 = 180
A+157 = 180
A = 180-157
A = 23
This is the measure of angle CAB.
Then we have one last set of steps.
(angle CAB)+(angle CAT) = angle BAT
(23)+(angle CAT) = 63
angle CAT = 63-23
angle CAT = 40 is the final answer.
My response is a bit long-winded. Another tutor may provide a much more efficient pathway.
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