Question

triangle abc is shown below. what is the length of line segment ac? 7 9 14 18

Explanation:

Step1: Identify triangle type

Since angles at B and C are equal, triangle ABC is isosceles with \( AB = AC \) or \( AB = BC \) or \( AC = BC \). From sides, \( AB = 2x \), \( AC = 3x - 7 \), \( BC = 4x - 10 \). Equal angles imply \( AB = AC \)? Wait, no, equal base angles imply \( AB = AC \)? Wait, angle at B and angle at C are equal, so sides opposite are equal. Side opposite angle B is \( AC \), side opposite angle C is \( AB \). So \( AB = AC \)? Wait, no: in triangle, side opposite angle B is \( AC \), side opposite angle C is \( AB \). So if \( \angle B=\angle C \), then \( AC = AB \). Wait, \( AB = 2x \), \( AC = 3x - 7 \). So \( 2x = 3x - 7 \)? Wait, no, maybe \( AB = BC \) or \( AC = BC \). Wait, let's check: if \( \angle B=\angle C \), then sides opposite are equal. So side opposite \( \angle B \) is \( AC \), side opposite \( \angle C \) is \( AB \). So \( AC = AB \)? Then \( 2x = 3x - 7 \), solving \( x = 7 \). Then \( AB = 14 \), \( AC = 14 \), \( BC = 4*7 -10 = 18 \). But wait, maybe \( AB = BC \). Let's try \( AB = BC \): \( 2x = 4x -10 \), \( 2x =10 \), \( x=5 \). Then \( AC = 3*5 -7 = 8 \), not in options. If \( AC = BC \): \( 3x -7 = 4x -10 \), \( x=3 \). Then \( AB = 6 \), \( AC = 2 \), \( BC = 2 \), not valid. So correct is \( AB = AC \), \( x=7 \). Then \( AC = 3*7 -7 = 14 \). Wait, but let's verify. If \( x=7 \), \( AB =14 \), \( AC=14 \), \( BC=18 \). Triangle with sides 14,14,18: valid. So length of AC is 14.

Step2: Solve for x

From \( AB = AC \) (since \( \angle B = \angle C \), so sides opposite are equal: \( AB = AC \)):
\( 2x = 3x - 7 \)
Subtract \( 2x \) from both sides:
\( 0 = x - 7 \)
So \( x = 7 \)

Step3: Calculate AC

Substitute \( x = 7 \) into \( AC = 3x - 7 \):
\( AC = 3(7) - 7 = 21 - 7 = 14 \)

Answer:

14