How Do You Calculate the Length of TP in a Common Tangents Problem?

[guan721]

The diagram shows two circles with a common tangent at T. The radius of the smaller circle is 5 cm and BT=BP. ACBP is a tangent to the circle with centre O. Calculate
(a) The length of TP

My attempt
- I tried to solve the question by finding complimentary angle of angle X, which is 90 degree - 32.5 degree(angle X) = 57.5 degree, and by using Trigonometry to calculate half the length of TP, Tan 57.5 degree multiply 5 cm, = 7.85 cm , and then multiply by 2, 7.85 cm x 2, the answer from me is 15.70 cm, but the correct answer is 17.15 cm.

Thanks a lot.

 

[PetSounds]

Your diagram is small and rather pockmarked. What does the 65° marking refer to? It would also help if you showed the process by which you calculated x.

 

[Mark44]

The diagram shows two circles with a common tangent at T. The radius of the smaller circle is 5 cm and BT=BP. ACBP is a tangent to the circle with centre O. Calculate
(a) The length of TP

My attempt
- I tried to solve the question by finding complimentary angle of angle X, which is 90 degree - 32.5 degree(angle X) = 57.5 degree, and by using Trigonometry to calculate half the length of TP, Tan 57.5 degree multiply 5 cm, = 7.85 cm , and then multiply by 2, 7.85 cm x 2, the answer from me is 15.70 cm, but the correct answer is 17.15 cm.

Where did 32.5° come from? Obviously, that's half of 65°, but it appears from your picture that the angle made by segment AT and the left edge of the horizontal line is 65°. Are you assuming that angle ATB is a right angle? This wouldn't be true unless point C happens to be the center of the larger circle.

 

[guan721]

Where did 32.5° come from? Obviously, that's half of 65°, but it appears from your picture that the angle made by segment AT and the left edge of the horizontal line is 65°. Are you assuming that angle ATB is a right angle? This wouldn't be true unless point C happens to be the center of the larger circle.

I calculate value x by following steps:
1) From the principle of circle, the angle formed by the tangent and the chord is equal to the angle in the alternate segment which is subtended by the chord
2) Angle formed by Tangent TP, and the Chord TB, is 65 degree, equal to the angle in the alternate segment, subtended by Chord TB, which is angle CBT
3) Thus angle CBT is 65 degree
4) Since line ACBP is a straight line, angle TBP consider as complimentary angle of angle CBT, equal to 180 degree - 65 degree = 115 degree
5) Since line TB and line PB are equal in length, we consider triangle TBP is Isosceles triangle
6) Angle x = (180 degree - 115 degree) divide by 2 = 32.5 degree

 

[Mark44]

I calculate value x by following steps:
1) From the principle of circle, the angle formed by the tangent and the chord is equal to the angle in the alternate segment which is subtended by the chord

This might be right, but I'm not following what you're saying. Rather than describing the angles and segments in words, describe them using the given points; for example, as chord BT or ∠ABT.
What "principle of circle" are you talking about?
"the angle formed by the tangent and the chord" -- ∠PTB, right?
" the angle in the alternate segment which is subtended by the chord" -- you need to identify this better. I don't know which segment you mean by "alternate segment". Please identify the segments using the points identified in the image.
Are you talking about angle y?