Question 954475: Dear Tutor,
So, I am in Geometry, and we are starting trig. The question is "Points A,B,C, and D are consecutive vertices of a regular decagon with sides 20 cm long. Ray AB and Ray DC are drawn and intersect at X. Find BX". I have drawn the problem out, and I get that it forms a 90 degree angle, but I can't figure out the rest. Please help me! And thank you for taking your time to help me!
-Shannon
Found 3 solutions by Alan3354, josgarithmetic, macston:
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! "Points A,B,C, and D are consecutive vertices of a regular decagon with sides 20 cm long. Ray AB and Ray DC are drawn and intersect at X. Find BX".
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Each exterior angle is 36 degrees (360/10)
CBX is an isosceles triangle with angles 36, 36 & 108 degrees.
The altitude from X to BC (Point Y) makes 2 right triangles with angles of 36, 54 & 90 degrees and BY = 1/2 of BC = 10
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cos(36) = 10/BX
BX = 10/cos(36)
BX =~ 12.36
Answer by josgarithmetic(39629)  (Show Source):
You can put this solution on YOUR website! Not being a finished answer, this may help:
Your 10-sided polygon, regular, will contain 8 triangles, each work 180 degrees, and has 10 vertices. EACH vertex on the interior has  degrees.
Your other, outer triangle AXB, is an isosceles triangle. ...
Can you make the drawing of all this, and find the two equal interior angles of the ISOSCELES TRIANGLE, AXB?
Answer by macston(5194)  (Show Source):
You can put this solution on YOUR website! For a decagon (n=no. of sides=10)
 =interior angle
 =144 degrees
Interior and exterior angles are supplementary so exterior angles are 36 degrees.
Angle XBD and Angle XCB are each 36 degrees, so Angle BXC is the third angle of the triangle =180-72=108 degrees
From the midpoint of side BC to X is one leg of a right triangle, from midpoint to B is the other leg and BX is hypotenuse. Length of B to midpoint is (1/2)20cm=10cm so cos(36)=10cm/BX, BX=10cm/cos(36)=12.36cm
ANSWER: BX=12.36 cm
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