a What can you tell about an inscribed triangle if one of its sides is a diameter of the circle?
B
b Draw chord GK and notice that diameter FH is perpendicular to this chord.
A
a See solution.
B
b FJ=6 inches
JH=2 inches
GJ=2sqrt(3) inches
GK=4sqrt(3) inches
Practice makes perfect
a Having been given a few points labeled on the cutting board, we want to write a proportion relating GJ and JH.
Note that FH is a diameter of the cutting board.
By the Inscribed Right Triangle Theorem, we know that if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. Therefore, ∠FGH=90 ^(∘) and △FGH is a right triangle.
FH is the hypotenuse of △FGH. We can also see that GJ is an altitude drawn to the hypotenuse.
With this information, we can find a proportion relating GJ and JH using the Right Triangle Similarity Theorem, Theorem 9.6.
Right Triangle Similarity Theorem
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
In our case, it means that △FGJ ~ △GHJ, △FGJ ~ △FHG, and △FHG ~ △GHJ.
Knowing this, we can write a proportion relating GJ and JH.
FJ/GJ=GJ/JH
b We want to find FJ, JH, and GJ. Recall that every board used to make the circular cutting board is 1-inch. We can see that FJ is equal to the width of six boards, so FJ=6. Similarly, we can conclude that JH=2. Let's substitute these values into the proportion obtained in Part A.
FJ/GJ=GJ/JH ⇒ 6/GJ=GJ/2Using this equality, we can calculate GJ.
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
We are given that ∠ FJG=90 ^(∘). Therefore, by this theorem we know that diameter FG bisects chord GK.
Knowing that GK is twice GJ, we can finally calculate GK.
GK = 2 GJ ⇒ GK&=2( 2sqrt(3))
&= 4sqrt(3)
