Question 1168530: A regular octagon is emclosed with a circle.it has an extension piece CDFE obtained by extendind F and C to E with BCE is a straight line.AF was the diameter of the circle. FE is a tangent to the circle at F.calclate the size of z(angle FEC)
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this geometry problem step by step.
**1. Diagram and Definitions**
* We have a regular octagon inscribed in a circle.
* AF is a diameter of the circle.
* FC is extended to E, such that BCE is a straight line.
* FE is a tangent to the circle at F.
* We need to find the measure of angle FEC.
**2. Properties of a Regular Octagon**
* A regular octagon has 8 equal sides and 8 equal interior angles.
* The measure of each interior angle of a regular octagon is:
* (n - 2) * 180° / n = (8 - 2) * 180° / 8 = 6 * 180° / 8 = 135°
**3. Angles in the Diagram**
* Let O be the center of the circle.
* Since the octagon is regular, the central angle subtended by each side is 360° / 8 = 45°.
* Angle AOF is 45 degrees.
* Angle COF is 45 degrees.
* Angle BOC is 45 degrees.
* Since AF is a diameter, angle ACF is 135/2 = 67.5 degrees, because the angle ACF is half the interior angle of the octagon.
* Angle AFC = 135/2 = 67.5 degrees.
**4. Tangent and Diameter Properties**
* Since FE is a tangent to the circle at F, angle AFE = 90°.
**5. Finding Angle CFE**
* We need to find angle CFE.
* Since angle AFC = 67.5 degrees and angle AFE = 90 degrees.
* angle CFE = angle AFE - angle AFC = 90 - 67.5 = 22.5 degrees.
**6. Finding Angle FEC**
* In triangle FEC, we need to find angle FEC.
* Since BCE is a straight line, angle FCB = 180 - 135 = 45 degrees.
* We know angle CFE = 22.5 degrees.
* In triangle FEC, the sum of the angles is 180 degrees.
* angle FEC + angle CFE + angle FCE = 180 degrees.
* angle FEC + 22.5 + 45 = 180
* angle FEC + 67.5 = 180
* angle FEC = 180 - 67.5
* angle FEC = 112.5 degrees.
**Therefore, the size of angle FEC is 112.5 degrees.**
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
A regular octagon is emclosed with a circle.it has an extension piece CDFE obtained by extendind
F and C to E with BCE is a straight line.AF was the diameter of the circle. FE is a tangent to the circle at F.
calclate the size of z(angle FEC)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The way how the problem is presented in the post, is  .
It is not a way presenting Math problem.
Throw this composition to a garbage bin, where its right place is.
|
|
|
