Answered

Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

[tex]$\overline{YS}$[/tex] is the perpendicular bisector of [tex]$\Delta XYZ$[/tex], and [tex]$\overline{YS}$[/tex] is a shared side of [tex]$\triangle XYS$[/tex] and [tex]$\triangle ZYS$[/tex]. Which of the following must be congruent in order to verify that [tex]$\triangle XYS \cong \triangle ZYS$[/tex]?

A. [tex]$\overline{ZY} \cong \overline{ZS}$[/tex]
B. [tex]$\overline{XY} \cong \overline{YS}$[/tex]
C. [tex]$\overline{XS} \cong \overline{ZS}$[/tex]
D. [tex]$\overline{XS} \cong \overline{XY}$[/tex]

Sagot :

GinnyAnswer
To verify that [tex]\(\triangle XYS \cong \triangle ZYS\)[/tex], we need to use congruence criteria such as SAS (Side-Angle-Side), ASA (Angle-Side-Angle), or SSS (Side-Side-Side). Since [tex]\(\overline{YS}\)[/tex] is already given to be a shared side, we need to detail further steps to check for congruence.

### Given Information:
- [tex]\(\overline{YS}\)[/tex] is the perpendicular bisector of [tex]\(\overline{XZ}\)[/tex].
- [tex]\(\overline{YS}\)[/tex] is a common side of both triangles [tex]\(\triangle XYS\)[/tex] and [tex]\(\triangle ZYS\)[/tex].

### Steps to Solve:

1. Perpendicular Bisector Property:
Since [tex]\(\overline{YS}\)[/tex] is the perpendicular bisector of [tex]\(\overline{XZ}\)[/tex], it implies two things:
- [tex]\(\overline{XS} \cong \overline{ZS}\)[/tex]
- [tex]\(\angle YSX \cong \angle YSZ = 90^\circ\)[/tex]

2. Common Side:
Both triangles share side [tex]\(\overline{YS}\)[/tex].

3. Congruence Criteria:
To show [tex]\(\triangle XYS \cong \triangle ZYS\)[/tex], we consider the Side-Angle-Side (SAS) criterion:
- Side 1: [tex]\(\overline{YS}\)[/tex] (common to both triangles)
- Angle: [tex]\(\angle YSX \cong \angle YSZ\)[/tex] (both right angles from the perpendicular bisector)
- Side 2: [tex]\(\overline{XS} \cong \overline{ZS}\)[/tex] (since [tex]\(\overline{YS}\)[/tex] is the perpendicular bisector of [tex]\(\overline{XZ}\)[/tex])

From the given options:

- Option A: [tex]\(\overline{ZY} \cong \overline{ZS}\)[/tex] is not implied by the given conditions.
- Option B: [tex]\(\overline{XY} \cong \overline{YS}\)[/tex] is not necessarily true and not required by the given conditions.
- Option C: [tex]\(\overline{XS} \cong \overline{ZS}\)[/tex] is true as [tex]\(\overline{YS}\)[/tex] is the perpendicular bisector of [tex]\(\overline{XZ}\)[/tex].
- Option D: [tex]\(\overline{XS} \simeq \overline{XY}\)[/tex] is irrelevant and incorrect.

### Answer:
The sides that must be congruent in order to verify that [tex]\(\triangle XYS \cong \triangle ZYS\)[/tex] are:
[tex]\[ \boxed{\overline{X S} \cong \overline{Z S}}. \][/tex]
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.