X represents which point of concurrency? O A. circumcenter O B. incenter C. orthocenter O D. centroid

Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
icon
Related questions
icon
Concept explainers
Topic Video
Question

Question 15

### Problem Description

#### Diagram:

A triangle \( \triangle RST \) is depicted. Point \( R \) is at the top vertex, \( T \) at the bottom-left vertex, and \( S \) at the bottom-right vertex. 

Within the triangle, several internal lines are drawn:
- Segment \( RU \) extends from vertex \( R \) to point \( U \) on side \( TS \).
- Segment \( RV \) extends from vertex \( R \) to point \( V \) on side \( ST \).
- Segment \( XW \) extends from point \( X \) (near the centroid area) to point \( W \) on side \( TS \).

Two line segments \( XU \) and \( XW \) are given with the following equations:
- \( XU = -2y + 15 \)
- \( XW = 3y + 5 \)

#### Question:

The problem asks to determine which point of concurrency \( X \) represents. The options are:
- **A. Circumcenter**
- **B. Incenter**
- **C. Orthocenter**
- **D. Centroid**

### Explanation of Diagrams:

The given triangle shows several angular and segment markings which indicate it involves concurrency points:
- \( X \) is suggested to be a notable center of the triangle due to concurrent lines.
- The exact nature of \( X \) is to be determined from the provided equations and geometric properties.

### Question Analysis:

The points of concurrency in a triangle are:
- **Circumcenter**: The point where the perpendicular bisectors of the sides intersect.
- **Incenter**: The point where the angle bisectors intersect.
- **Orthocenter**: The point where the altitudes intersect.
- **Centroid**: The point where the medians intersect; it is also the centroid that divides each median into a ratio of 2:1.

Given \( XU \) and \( XW \) are mid-segmental evaluations, determining their congruence will ensure identifying the nature of \( X \).

### Answer Choices:

Which point of concurrency does \( X \) represent?

- **A. Circumcenter**
- **B. Incenter**
- **C. Orthocenter**
- **D. Centroid**
Transcribed Image Text:### Problem Description #### Diagram: A triangle \( \triangle RST \) is depicted. Point \( R \) is at the top vertex, \( T \) at the bottom-left vertex, and \( S \) at the bottom-right vertex. Within the triangle, several internal lines are drawn: - Segment \( RU \) extends from vertex \( R \) to point \( U \) on side \( TS \). - Segment \( RV \) extends from vertex \( R \) to point \( V \) on side \( ST \). - Segment \( XW \) extends from point \( X \) (near the centroid area) to point \( W \) on side \( TS \). Two line segments \( XU \) and \( XW \) are given with the following equations: - \( XU = -2y + 15 \) - \( XW = 3y + 5 \) #### Question: The problem asks to determine which point of concurrency \( X \) represents. The options are: - **A. Circumcenter** - **B. Incenter** - **C. Orthocenter** - **D. Centroid** ### Explanation of Diagrams: The given triangle shows several angular and segment markings which indicate it involves concurrency points: - \( X \) is suggested to be a notable center of the triangle due to concurrent lines. - The exact nature of \( X \) is to be determined from the provided equations and geometric properties. ### Question Analysis: The points of concurrency in a triangle are: - **Circumcenter**: The point where the perpendicular bisectors of the sides intersect. - **Incenter**: The point where the angle bisectors intersect. - **Orthocenter**: The point where the altitudes intersect. - **Centroid**: The point where the medians intersect; it is also the centroid that divides each median into a ratio of 2:1. Given \( XU \) and \( XW \) are mid-segmental evaluations, determining their congruence will ensure identifying the nature of \( X \). ### Answer Choices: Which point of concurrency does \( X \) represent? - **A. Circumcenter** - **B. Incenter** - **C. Orthocenter** - **D. Centroid**
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

Blurred answer
Knowledge Booster
Sample space, Events, and Basic Rules of Probability
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning