Geometry word problem about circles, tangents, and triangles.

Question

You have a circle and 2 tangents to the circle, AB and CB, with A and C being the points on the circle and B being the point where the 2 tangents intersect. AB and CB equal 16cm. D is any point on the minor arc AC, and a tangent to the circle is drawn at D. This tangent intersects AB at point E and CB at point F. What is the perimeter of triangle EFB?

Answer 1

Call the center of the circle O.

 

Look at triangles AEO and DEO.  EO is a common side of both triangles.  AO is equal to the radius of the circle, as is DO.  Angles EAO and EDO are right angles, because a tangent is perpendicular to the line segment from the tangent point to the center.  By the SSA "hypotenuse-leg" criterion, these two triangles are congruent.

 

Therefore the length of AE is equal to the length of ED.

 

Similarly you can show that the length of DF is equal to the length of FC.

 

Now look at ΔEFB.  Its perimeter is equal to EB + FB + EF.  EF is equal to ED + DF.  But ED = AE, and DF = FC.

 

So EF = AE + FC, and therefore the perimeter of the triangle is EB + FB + AE + FC.  Rearrange the terms and this is EB + AE + FB + FC.

 

But EB + AE = AB = 16 cm.  And FB + FC = CB = 16 cm.

 

So the perimeter of the triangle is 16 + 16 = 32 cm.