Answer:
CF = 12
DE = 23
CD = 23
DF = 23.9
Explanation:
We can see that there are 2 right-angled triangles: DEF and CDF with common side FD.
They are right angled triangles because the EF and CF are radii of the circle and these segments intersect the tangents at 90°
The common side DF is the hypotenuse for both triangles
The sides EF and CF are radii of the circle so EF = CF
We have two triangles which have two sides equal and one of the angles equal to the corresponding angle of the other
By the SSA theorem, the two triangles are similar
SSA Theorem
if two sides and an angle not included between them are respectively equal to two sides and an angle of the other then the two triangles are equal
Therefore by the law of similar triangles,
CD = EF
Plugging in the expressions we get
13x - 16 = 4x + 11
Subtract 4x from both sides:
13x - 4x - 16 = 11
9x - 16 = 11
Add 16 to both sides:
9x - 16 + 16 = 11 + 16
9x = 27
x = 27/9 = 3
Therefore
ED = 4x + 11 = 4(3) + 11 = 12 + 11 = 23
and this is equal to CD
Using the Pythagorean theorem for right triangles,
For ΔDEF,
DF² = EF² + DE²
DF² = 12² + 23²
= 144 + 529
= 673
DF = √673
= 25.9422
= 25.9 rounded to the nearest tenth
So the measures of the segments are
CF = 12
DE = 23
CD = 23
DF = 23.9