CPM Homework Help : CCG Problem 10-107

Review circle relationships as you answer the questions below.

On your paper, draw a diagram of $⊙B$ with $\overarc{ A C }$ . If $m\overarc{AC}=80°$ and the length of the radius of $⊙B$ is $10$ , find the length of chord $\overline{ A C }$ .

Hint (a):

More Help (a):

According to the Law of Cosines,
$b^2=a^2+c^2−2ab\cos\left(\text{B}\right)$
Therefore, $b^2=10^2+10-2\left(10\right)\left(10\right)\cos\left(80°\right)$

Circle with center, B, and points, a, and c, line segments from, B, to C, from, B, to, A, from, A, to, C, with labels as follows: Arc, A, C, 80 degrees, angle, a, b, c, 80 degrees, segment, A, b, and B, C, each labeled 10.

Now draw a diagram of a circle with two chords, $\overline{EF}$ and $\overline{GH}$ , that intersect at point $K$ . If $EF=15$ , $EK=6$ , and $HK=3$ , what is $GK$ ?

Hint (b):

More Help (b):

Remember that intersecting chords create similar triangles, and that the corresponding sides of similar triangles have equal ratios.

Step (b):

$\frac{9}{3}=\frac{x}{6}$

Circle with points, in order, g, F, H, E, line segments from, e, to, g, from, e, to, f, from, h, to, g, from, h, to, f. Intersection of segments, g, h, and, e, f, labeled, k. Segment, e, k, labeled 6, segment, k, f, labeled 9, Segment, k, h, labeled 3.