**Title: Orthogonality in a Circle with Diameter****Objective:**To demonstrate that the line segments \( \overrightarrow{CA} \) and \( \overrightarrow{CB} \) are orthogonal without using coordinate geometry.**Problem Statement:**Suppose that \( AB \) is the diameter of a circle with center \( O \), and that \( C \) is a point on one of the two arcs joining \( A \) and \( B \). Prove that \( \overrightarrow{CA} \) and \( \overrightarrow{CB} \) are orthogonal.**Diagram Explanation:**1. The diagram shows a circle with center \( O \) and diameter \( AB \).2. Point \( A \) is positioned on the left end of the diameter, and point \( B \) is on the right end.3. There is a vector \( \overrightarrow{u} \) originating from the center \( O \) to \( B \), and \( -\overrightarrow{u} \) from \( O \) to \( A \).4. Point \( C \) is marked on the upper arc of the circle, above the diameter \( AB \).5. Vector \( \overrightarrow{v} \) points from \( O \) to \( C \).**Proof Approach Without Coordinates:**1. **Central Angle Subtends a Right Angle:** - In any circle, an angle subtended by the diameter (i.e., angle \( \angle ACB \)) is a right angle (90 degrees). This is a well-known result from the properties of a circle. 2. **Orthogonality:** - Two vectors (or line segments in this context) are orthogonal if the angle between them is 90 degrees. - Here, \( \angle ACB = 90^\circ \), implying that \( \overrightarrow{CA} \) and \( \overrightarrow{CB} \) form a right angle with each other.Based on these geometric principles, we can conclude that vectors \( \overrightarrow{CA} \) and \( \overrightarrow{CB} \) are indeed orthogonal.

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Description: **Title: Orthogonality in a Circle with Diameter****Objective:**To demonstrate that the line segments \( \overrightarrow{CA} \) and \( \overrightarrow{CB} \) are orthogonal without using coordinate geometry.**Problem Statement:**Suppose that \( AB \

given: AB is the diameter of the circle with center O and that C is a point on one of the two arcs…

Answered: Suppose that AB is the diameter of a…

Math

Algebra

Suppose that AB is the diameter of a circle with çenter O and that C is a point on one of the two arcs joining A and B. Show that CÁ and CB are orthogonal without using coordinates. A В

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter10: Analytic Geometry

Section10.4: Analytic Proofs

Problem 19E: Use the analytic method to decide what type of triangle is formed when the midpoints of the sides of...

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