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I am very confused by these problems
1

2.

 Mar 7, 2024
 #1
avatar+128406 
+1

First one

Let the center  of the smaller circle  = M   and the larger one, N

Draw MA and NB

Extend   NM  and BA  to meet at  C

Let MC   =  3 + x

 

We have similar triangles  CMA and CNB

So

CM / MA = CN / NB

 

(3 + x) / 3   = (13 + x) / 7

7(3 + x) = 3(13 + x)

21 + 7x  = 39 + 3x

4x   = 18

x = 18/4 = 9/2  =  4.5

 

And we  can use similar triangles again  to  find TP

 

CM / MA  =  CT / TP

(3 + 4.5) / 3 = (4.5 + 6)  / TP

TP  = 3(4.5 + 6) / (3 + 4.5)

TP  = 3 (10.5) / ( 7.5)  =  4.2

 

cool cool cool

 Mar 7, 2024
 #2
avatar+128406 
+1

Second one

Law of Cosines

AC^2  = AB^2 + BC^2  - 2(AB * BC) cos ABC

30^2  = 25^2 + 25^2  - 2(25 * 25) cos ABC

[ 30^2 - 25^2 -25^2 ] /  (-2 * 25 * 25)   = cos ABC

cos ABC =  7/25

 

sin ABC   = sqrt [ 25^2  - 7^2 ] / 25 =  sqrt [ 576 ] / 25  =   24 / 25

 

cool cool cool

 Mar 7, 2024

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