Question

triangle abc is shown below. what is the length of line segment ac? 7 9 14 18 (diagram: triangle abc with a at top, b and c at base; ab labeled 2x, ac labeled 3x−7, bc labeled 4x−10; angles at b and c marked equal)

Explanation:

Step1: Identify triangle type (isosceles)

Since angles at B and C are equal, \( AB = AC \)? Wait, no, wait—wait, in a triangle, if two angles are equal, the sides opposite are equal. Wait, the angles at B and C: so side opposite angle B is AC, side opposite angle C is AB? Wait, no, angle at B: opposite side is AC; angle at C: opposite side is AB. Wait, but in the diagram, AB is \( 2x \), AC is \( 3x - 7 \), and BC is \( 4x - 10 \). Wait, maybe it's an isosceles triangle with \( AB = AC \)? Wait, no, maybe \( AB = BC \)? Wait, no, the angles at B and C are marked equal, so triangle ABC is isosceles with \( AB = AC \)? Wait, no, angle at B and angle at C: so sides opposite are AC and AB. Wait, no, angle at B: opposite side is AC; angle at C: opposite side is AB. So if angles at B and C are equal, then \( AC = AB \)? Wait, no, \( AB \) is length \( 2x \), \( AC \) is \( 3x - 7 \), \( BC \) is \( 4x - 10 \). Wait, maybe I made a mistake. Wait, maybe \( AB = BC \)? No, let's check again. Wait, in triangle ABC, angles at B and C are equal, so it's isosceles with \( AB = AC \)? Wait, no, \( AB \) is from A to B: length \( 2x \), \( AC \) is from A to C: \( 3x - 7 \), \( BC \) is from B to C: \( 4x - 10 \). Wait, maybe the equal angles mean \( AB = AC \)? Wait, no, if angles at B and C are equal, then sides opposite are equal. Side opposite angle B is AC, side opposite angle C is AB. So \( AC = AB \)? Wait, \( AB = 2x \), \( AC = 3x - 7 \). So \( 2x = 3x - 7 \)? Solving that: \( 2x = 3x - 7 \) → \( -x = -7 \) → \( x = 7 \). Then \( AC = 3(7) - 7 = 14 \). Wait, but let's check BC: \( 4x - 10 = 4(7) - 10 = 18 \). Wait, but if \( AB = 2x = 14 \), \( AC = 14 \), \( BC = 18 \), that works. Alternatively, maybe \( AB = BC \)? Let's try that. \( 2x = 4x - 10 \) → \( -2x = -10 \) → \( x = 5 \). Then \( AC = 3(5) - 7 = 8 \), which is not an option. Or \( AC = BC \): \( 3x - 7 = 4x - 10 \) → \( -x = -3 \) → \( x = 3 \). Then \( AB = 6 \), \( AC = 2 \), \( BC = 2 \), which is invalid (triangle inequality). So the correct one is \( AB = AC \), \( x = 7 \), so \( AC = 14 \).

Step1: Determine triangle is isosceles (∠B=∠C ⇒ AB=AC)

Given \( AB = 2x \), \( AC = 3x - 7 \). So \( 2x = 3x - 7 \).

Step2: Solve for \( x \)

\( 2x = 3x - 7 \)
Subtract \( 2x \) from both sides: \( 0 = x - 7 \)
Add 7 to both sides: \( x = 7 \).

Step3: Calculate \( AC \)

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

Answer:

14