Question

triangle congruence: sss and hl
point h is the midpoint of side fk.
for the triangles to be congruent by sss, what must be the value of x?
(there is a diagram with triangle fhg and fhj, with hg and hj marked as equal, fg = 3x - 2, fj = 2x + 4, and four boxes with numbers 3, 8, 6, 1 below the diagram)

Explanation:

Step1: Identify congruent sides

Since triangles \( \triangle FHG \) and \( \triangle KHJ \) are congruent by SSS, their corresponding sides \( FG \) and \( KJ \) must be equal. So, \( 3x - 2 = 2x + 4 \).

Step2: Solve for \( x \)

Subtract \( 2x \) from both sides: \( 3x - 2x - 2 = 2x - 2x + 4 \), which simplifies to \( x - 2 = 4 \). Then add 2 to both sides: \( x - 2 + 2 = 4 + 2 \), so \( x = 6 \). Wait, but let's check again. Wait, maybe I misread. Wait, the sides \( FG = 3x - 2 \) and \( KJ = 2x + 4 \). For SSS congruence, these sides should be equal. So:
\( 3x - 2 = 2x + 4 \)
Subtract \( 2x \) from both sides: \( x - 2 = 4 \)
Add 2 to both sides: \( x = 6 \)? Wait, but the options are 3,8,6,1? Wait, the user's image has options 3,8,6,1? Wait, maybe I made a mistake. Wait, let's re-express. Wait, maybe the sides are \( FH = HK \) (since H is midpoint), \( HG = HJ \) (marked equal), so \( FG = KJ \). So \( 3x - 2 = 2x + 4 \). Solving: \( 3x - 2x = 4 + 2 \) → \( x = 6 \). Wait, but the options include 6? Wait, the user's image shows options 3,8,6,1? So the correct \( x \) is 6? Wait, maybe I miscalculated. Wait, \( 3x - 2 = 2x + 4 \) → \( x = 6 \). Yes.

Answer:

6