Can someone help me please?
- NeilLv 57 years agoFavorite Answer
OK this might be a little hard to explain without being able to use lots of pictures so I hope you can follow:
1) Where the points of the small triangle touch the sides of the big triangle three new triangles are created: DAB, BCF and AEC.
2) Because the sides of these three triangles that are outside the circle are all tangents to the circle the two angles of each that touch the circle are equal (Ie: All three are isosceles triangles) so for example in triangle AEC angle A and angle C are equal.
3) From this we can start to fill in some more of the angles (This is where it gets hard to explain without lots of pictures!) Firstly triangle AEC we know 'angle E' = 56 and 'angle A' = 'angle C' so since the internal angles of a triangle = 180 we can work out A and C.
A + C + E = 180 [We are told E = 56]
A + C + 56 = 180 [Rearrange]
A + C = 180 - 56 [Simplify]
A + C = 124 [A = C so each equal 124/2]
A and C both equal 62
4) Next if we look at the line EF we know the angles on a straight line add up to 180, at point C this angle is split into three, the first we are given as 38, the second we are given as x and the third we have just calculated to be 62, so:
38 + 62 + x = 180
100 + x = 180
x = 80
5) Now look at the triangle BCF, we are told C = 38, we know since the two sides CF and BF are tangents to the circle that angle B = angle C, so if C = 38 then B = 38, therefore:
F + B + C = 180 [B and C both equal 38]
F + 38 + 38 = 180
F + 76 = 180
F = 180 - 76
F = 104
6) Now look at the original bigger triangle DEF, we now know E = 56 and F = 104, therefore:
D + E + F = 180
D + 56 + 104 = 180
D + 160 = 180
D = 180 - 160
D = 20
*Point D is angle Y in the question, so Y = 20.
7) Now look at triangle DAB, we know because it forms tangents A = B and we know D = 20, so:
A + B + D = 180
A + B + 20 = 180
A + B = 180 - 20
A + B = 160
Since A = B, A and B each equal 160 / 2 which equals 80.
8) Next look at point A, again angles along a straight line so totaling 180, of which one we have just calculated to be 80, one we are given as z and one we calculated in step 3 to be 62, so:
80 + z + 62 = 180
142 + z = 180
z = 180 - 142
z = 38
9) Just for the sake of completion we might as well get the last angle of the inner triangle ABC, we know A (z) is 38 and C (x) is 80 so the missing point B is:
A + B + C = 180
38 + B + 80 = 180
118 + B = 180
B = 180 - 118
B = 62
There every angle filled in :)Source(s): x = 80 y = 20 z = 38