Question

Please Help I Don't Understand

Answer 1

In general, they're not similar.

In any triangle with side lengths a, b, and c, we have the aptly-named triangle inequality that says the largest side is no larger than the sum of the smaller sides. In other words, if a and b are both smaller than c, then

a + b ≥ c

Suppose x < 12. Then BC corresponds to either YZ or XY.

• If BC corresponds to YZ, then the triangles are similar if and only if

BC/YZ = AB/XY = AC/XZ

x/2 = 9/3 = 12/4 = 3   ⇒   x = 6

• If BC corresponds to XY, then triangle similarity means

BC/XY = AB/YZ = AC/XZ

x/3 = 9/2 = 12/4

but this fails because 9/2 ≠ 12/4 = 3.

Suppose x > 12. Then BC corresponds to XZ, and

x/4 = 12/3 = 9/2

but this also fails because 12/3 = 4 ≠ 9/2.

(We ignore the case of x = 12 because that would make ∆ABC isosceles, and ∆XYZ certainly is not.)

So ∆ABC and ∆XYZ are similar only if x = 6. Under this condition, similarity would follow from the SSS similarity theorem.