Question

given that point \\(x\\) is the incenter of this triangle, which of the following statements is true?

• \\(\overline{dx} \cong \overline{ex}\\)
• \\(\overline{cx} \cong \overline{bx}\\)
• \\(\angle fxc \cong \angle fxb\\)
• \\(\angle fxc \cong \angle axe\\)

Explanation:

Identify the properties of an incenter

The problem states that point \(X\) is the incenter of \(\triangle ABC\). By definition, the incenter of a triangle is the point of intersection of the angle bisectors of the triangle.

Apply the Incenter Theorem

The Incenter Theorem states that the incenter of a triangle is equidistant from all three sides of the triangle.

Relate distances to the given segments

The segments representing the perpendicular distances from the incenter \(X\) to the sides of \(\triangle ABC\) are:

• \(\overline{DX}\) (perpendicular to side \(\overline{AB}\))
• \(\overline{EX}\) (perpendicular to side \(\overline{AC}\))
• \(\overline{FX}\) (perpendicular to side \(\overline{BC}\))

According to the theorem, these perpendicular segments are equal in length:

$$DX = EX = FX$$

Therefore, the segment \(\overline{DX}\) is congruent to \(\overline{EX}\):

$$\overline{DX} \cong \overline{EX}$$

Evaluate the given options

• \(\overline{DX} \cong \overline{EX}\): This is true because the incenter is equidistant from the sides.
• \(\overline{CX} \cong \overline{BX}\): This is not necessarily true; the incenter is not generally equidistant from the vertices.
• \(\angle FXC \cong \angle FXB\): These angles are not necessarily congruent.
• \(\angle FXC \cong \angle AXE\): These angles are not necessarily congruent.

Answer:

• (A) \(\overline{DX} \cong \overline{EX}\) (Correct answer)
• (B) \(\overline{CX} \cong \overline{BX}\)
• (C) \(\angle FXC \cong \angle FXB\)
• (D) \(\angle FXC \cong \angle AXE\)