Question

1. \\(\overleftrightarrow{ac}\\) is the perpendicular bisector of \\(\overline{gh}\\). determine the length of the following sides. (lesson 4.4) (1 point)

a. \\(\overline{gh}\\)

b. \\(\overline{ch}\\)

c. with the information shown above, we can also determine that \\(\overline{bc} \cong \overline{bc}\\). therefore, which of the following theorems affirms that \\(\delta gcb \cong \delta hcb\\)? choose two.

a side-side-side congruence.
b angle-side-angle congruence.
c side-side-angle congruence.
d side-angle-side congruence.

Explanation:

Identify perpendicular bisector properties

Using the Solving Linear Equations knowledge point
Since line \(\overleftrightarrow{AC}\) is the perpendicular bisector of segment \(\overline{GH}\), point \(B\) is the midpoint of \(\overline{GH}\), and \(\overleftrightarrow{AC} \perp \overline{GH}\).

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Apply Perpendicular Bisector Theorem

The Perpendicular Bisector Theorem states that any point on the perpendicular bisector of a segment is equidistant from the endpoints of that segment. Since point \(C\) lies on the perpendicular bisector \(\overleftrightarrow{AC}\) of \(\overline{GH}\), it must be equidistant from \(G\) and \(H\).

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Determine triangle congruence criteria

We want to prove \(\Delta GCB \cong \Delta HCB\). Let's list the corresponding congruent parts we have established:

1. \(\overline{GB} \cong \overline{HB}\) (since \(B\) is the midpoint of \(\overline{GH}\))
2. \(\angle GBC \cong \angle HBC\) (both are right angles, \(90^\circ\), because \(\overleftrightarrow{AC} \perp \overline{GH}\))
3. \(\overline{BC} \cong \overline{BC}\) (Reflexive Property of Congruence)
4. \(\overline{CG} \cong \overline{CH}\) (proven in the previous step)

Using these parts, we can affirm congruence using:

• Side-Angle-Side (SAS): using \(\overline{GB} \cong \overline{HB}\), \(\angle GBC \cong \angle HBC\), and \(\overline{BC} \cong \overline{BC}\).
• Side-Side-Side (SSS): using \(\overline{GB} \cong \overline{HB}\), \(\overline{BC} \cong \overline{BC}\), and \(\overline{CG} \cong \overline{CH}\).

Answer:

Question 7a

\(GH = 16\)

Question 7b

\(CH = 12\)

Question 7c

• A Side - Side - Side Congruence. (Correct answer)
• B Angle - Side - Angle Congruence.
• C Side - Side - Angle Congruence.
• D Side - Angle - Side Congruence. (Correct answer)