Question

question 7 of 50
the triangles are similar by:
triangle with sides labeled: ac=6, cc?=3, cb?=4, ab=15, cb=8, and a segment of length 5
options:

• the sss similarity theorem
• the hl similarity theorem
• the aa similarity postulate

Explanation:

Step1: Identify side ratios

First, find the ratios of corresponding sides. For the smaller triangle (let's say triangle with sides 3, 4, 5) and the larger triangle (sides 6, 8, 15? Wait, no, wait: Wait, the sides are: from the diagram, the smaller triangle has sides 3, 4, 5; the larger triangle has sides 6 (32), 8 (42), and 15? Wait, no, wait 3/6 = 1/2, 4/8 = 1/2, 5/10? Wait, no, maybe I misread. Wait, the larger triangle: AC is 6, CB is 8, AB is 15? No, wait the smaller triangle: one side is 3, another 4, and 5 (hypotenuse). Then the larger triangle: AC is 6 (32), CB is 8 (42), and the other side: wait, maybe the sides are 3, 4, 5 and 6, 8, 10? Wait, maybe the diagram has AB as 15? No, maybe I made a mistake. Wait, the SSS similarity theorem states that if the ratios of all three corresponding sides are equal, then the triangles are similar. Let's check the ratios:

Smaller triangle sides: 3, 4, 5

Larger triangle sides: 6 (32), 8 (42), and let's see the third side: if the smaller triangle's third side is 5, then the larger's should be 10 (5*2) to have ratio 2. Wait, but in the diagram, maybe the larger triangle's third side is 10? Wait, maybe the diagram has a typo, but the options include SSS. Wait, let's recalculate:

Wait, the smaller triangle: sides 3, 4, 5 (3-4-5 triangle, right triangle). The larger triangle: sides 6 (32), 8 (42), and 10 (52) – but in the diagram, maybe AB is 10? Wait, the user's diagram: A to C is 6, C to the midpoint? Wait, no, the diagram shows: point A, C, B. The smaller triangle is inside, with sides 3 (AC segment), 4 (CB segment), 5 (the middle side). Then the larger triangle: AC is 6 (32), CB is 8 (42), and the other side (from A to B) should be 10 (52) if SSS. Wait, but the option is SSS similarity theorem. So let's check the ratios:

3/6 = 1/2, 4/8 = 1/2, 5/10 = 1/2? Wait, maybe the third side is 10, not 15. Maybe a typo. So if all three sides are in ratio 1/2, then SSS similarity applies.

HL similarity is for right triangles (hypotenuse-leg), but here we have three sides. AA is angle-angle, but here we can check sides. So the SSS similarity theorem: if all three corresponding sides are proportional, then similar. So the ratios are 3/6 = 1/2, 4/8 = 1/2, 5/10 = 1/2 (assuming the third side is 10, maybe the diagram has a typo with 15). So the SSS similarity theorem applies.

Step2: Eliminate other options

HL similarity is for right triangles where hypotenuse and one leg are proportional, but here we have three sides, so HL is out. AA is angle-angle, but we are given sides, so SSS is the one with all three sides proportional. So the correct answer is the SSS similarity theorem.

Answer:

the SSS similarity theorem