Question

Triangle ABC is shown below.

Triangle A B C is shown. The length of side A B is 2 x, the length of side B C is 4 x minus 10, and the length of side A C is 3 x minus 7. Angle A B C and B C A are congruent.

What is the length of line segment AC?

7
9
14
18

Answer 1

In triangle ABC, we have the following side lengths:
- AB = \(2x\)
- BC = \(4x - 10\)
- AC = \(3x - 7\)

Given that angles ABC and BCA are congruent, we can conclude that triangle ABC is isosceles with sides adjacent to these angles (AB and AC) being equal in length. Therefore, we set the lengths of those sides equal to each other:

\[
AB = AC
\]

Substituting the expressions for AB and AC gives us:

\[
2x = 3x - 7
\]

To solve for \(x\), we first isolate \(x\):

\[
2x - 3x = -7
\]
\[
-x = -7
\]
\[
x = 7
\]

Now that we have found \(x\), we can determine the lengths of the sides of the triangle:

1. Length of side AB:
\[
AB = 2x = 2(7) = 14
\]

2. Length of side AC:
\[
AC = 3x - 7 = 3(7) - 7 = 21 - 7 = 14
\]

3. Length of side BC:
\[
BC = 4x - 10 = 4(7) - 10 = 28 - 10 = 18
\]

Now we verify that the triangle inequality holds for all sides. We have:
- \(AB + AC = 14 + 14 = 28 > 18 = BC\)
- \(AB + BC = 14 + 18 = 32 > 14 = AC\)
- \(AC + BC = 14 + 18 = 32 > 14 = AB\)

All conditions of the triangle inequality are satisfied.

Finally, since we were asked for the length of line segment AC:
The length of line segment AC is \(14\).

Thus, the answer is \(\boxed{14}\).