I don’t understand how to solve this. Please help!

  GH = 16; CH = 12

Step-by-step explanation:

First of all, you need to understand the meaning of "perpendicular bisector." It means that GH is divided into two equal parts by line AC, and that AC makes a right angle to GH.

The right angle is marked.

(a) The length of one of the halves of GH is marked as being 8 units long, so the other half will also be 8 units long. Of course, the length of GH is the sum of its two halves:

  GH = GB +BH = 8 + 8

  GH = 16

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(b) Triangles CBG and CBH share side CB, so have that length in common. They have equal lengths BG and BH because BC bisects GH. They have a right-angle at B in common, so can be considered congruent by SAS, the fact that two congruent sides have a congruent angle between them.

Since triangles CBG and CBH are congruent, their corresponding sides CG and CH are also congruent. Side CG is marked 12 units long, so CH will be 12 units long, also.

  CH = 12

You could shortcut all of the congruent triangle logic by recognizing that an altitude (CB) is a perpendicular bisector of the base (GH) if and only if the triangle is isosceles. The sides of an isosceles triangle are always congruent, so CG = CH = 12.

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In part (c), you're supposed to choose possible theorems for demonstrating the congruence of the triangles we described above.

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.