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How are the segments cutting the angles of the triangle?
What type of triangles are △ BGE and △ BGF
Can you find the size of AE?
Point of concurrency: Incenter
Type of segments: Angle bisectors
AAS Congruence Theorem, for example.
≈ 3.9 cm
The point of concurrency where the distance to each of the triangle's sides are equidistant, is called the incenter. Such a point is created by the angle bisectors to the triangle's three vertices.
By the Incenter Theorem, we know two things regarding the segments from point G to each of the triangle's sides.
Also, since the segments that create the incenter are angle bisectors, we know that BG cuts ∠ EBF into two equal halves. The two triangle's also share BG as a side which means it's congruent by the Reflexive Property of Congruence. Let's add all of this information to the triangle's at hand.
Now we have enough information to use both the HL congruence Theorem or the AAS Congruence Theorem to prove congruence.
In Part B, we showed that △ BGF≅ △ BGE. Since BF=3 we know that the corresponding side in △ BEG, namely BE, is also 3. With this information, we can determine AE as 7 cm.
Examining the diagram, we see that △ AEG is a right triangle with a known hypotenuse and leg. By substituting these measures into the Pythagorean Theorem, we can find the length of the triangles last leg, which also happens to be the radius of the circle.
The radius of the circle is about 3.9 cm.
