Regard triangleTRI and the following facts: Segment IR || Segment NA TA= 12 cm, RA= 3 cm, NT= 18 cm, AN= 22 (a)What theorem or postulate proves Triangle ANT similar to RIT?(b)Solve for NI by using the Side-Splitting Theorem.Show all work.(c)What is the scale factor for similar triangles ANT and RIT?(d)Calculate RI. Show your work.(e)What is the ratio of the areas of the two triangles? Show your work.

A) Using the side-splitter theorem we know that the sides that are intersected by the parallel lines are proportional.  It can also be shown that the angles are congruent due to the parallel segments cut by a transversal.
B) NI = 4.5
C) 4:5
D) RI = 27.5
E) 16:25

Explanation
We will set up a proportion for b:
AT/RT = NI/TI
12/15 = 18/x

Cross multiply:
12*x = 18*15
12x = 270

Divide both sides by 12:
12x/12 = 270/12
x = 22.5

Since TI = TN + NI, we have
22.5 = 18 + NI
-18       -18
4.5 = NI

C) The ratio of the sides of the smaller triangle to the larger rectangle are given by the ratio AT/RT, or 12/15 = 4/5.

D)  Setting up a proportion we have
AN/RI = AT/RT
22/RI = 12/15

Cross multiply:
22*15 = RI*12
330 = 12RI

Divide both sides by 12:
330/12 = 12RI/12
RI = 27.5

E)  The ratio of the areas will be the square of the ratio of the side lengths, since area is a square measurement:
(4/5)² = 4²/5² = 16/25