Question

Point H is the midpoint of side FK.

Triangles F G H and K J H are connected at point H. The lengths of sides F H and H K are congruent. The lengths of sides G H and J H are congruent. The length of side F G is 3 x minus 2 and the length of corresponding side J K is 2 x + 4.

For the triangles to be congruent by SSS, what must be the value of x?

1
3
6
8

Answer 1

To determine the value of \( x \) that makes triangles \( FGH \) and \( KJH \) congruent by the Side-Side-Side (SSS) postulate, we need to set the lengths of the corresponding sides equal to each other.

1. We'll denote the lengths:
- \( FH = HK \)
- \( GH = JH \)
- \( FG = JK \)

2. We know that the lengths of sides \( FG \) and \( JK \) are given as:
- \( FG = 3x - 2 \)
- \( JK = 2x + 4 \)

3. For the triangles to be congruent, we set \( FG \) equal to \( JK \):
\[
3x - 2 = 2x + 4
\]

4. Now, we'll solve for \( x \):
- Subtract \( 2x \) from both sides:
\[
3x - 2x - 2 = 4
\]
\[
x - 2 = 4
\]

- Add 2 to both sides:
\[
x = 6
\]

Thus, \( x \) must be \( 6 \) for the triangles \( FGH \) and \( KJH \) to be congruent by SSS.

**Answer: 6**