This geometry question is confusing me, how do I solve it? Please explain

The angle bisector theorem states that if a line divides an angle of a triangle into two congruent angles, then the line divides the opposite side into segments that are proportional to the lengths of the other two sides of the triangle.

In this case, SH bisects angle CSR and CI bisects angle RCS. This means that SH divides CS into two segments that are proportional to CH and HS, and CI divides CS into two segments that are proportional to IS and SR.

Let's use the angle bisector theorem to find the lengths of HR and IR.

First, let's find the length of HR:

Since SH bisects angle CSR, we can set up the following proportion:

CH / HS = CS / SR

Substituting the given values:

4 / HS = 8 / SR

Cross multiplying:

4 * SR = 8 * HS

Dividing both sides by 4:

SR = 2 * HS

Since we know that HS = 8 (given), we can substitute this value:

SR = 2 * 8

SR = 16

So, the length of SR is 16.

Since HR = HS + SR, we can substitute the given values:

HR = 8 + 16

HR = 24

Therefore, the length of HR is 24.

Now, let's find the length of IR:

Since CI bisects angle RCS, we can set up the following proportion:

IS / SR = CI / CR

Substituting the given values:

3 / SR = CI / CR

Cross multiplying:

3 * CR = SR * CI

Dividing both sides by 3:

CR = (SR * CI) / 3

Substituting the given values:

CR = (16 * CI) / 3

Therefore, the length of CR is (16 * CI) / 3.

Since IR = IS + CR, we can substitute the given values:

IR = 3 + (16 * CI) / 3

Therefore, the length of IR is 3 + (16 * CI) / 3.

In conclusion, the length of HR is 24 and the length of IR is 3 + (16 * CI) / 3.

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