Chapter 1: Vectors, Lines and Planes
Section 1.3: Dot Product
Use vector methods to prove that an angle inscribed in a semicircle is necessarily a right angle.
Angle PRQ is inscribed in the semicircle shown in Figure 1.3.12(a). Vectors A and −A are along the diagonal, and vector B, connecting points O (the center of the circle) and R, is along a radius.
Vectors A and B have length r, the length of the radius of the circle whose center is O.
The vectors B+A and B−A are along the line segments PR and QR, respectively. The angle between these two vectors is ∠PRQ.
The cosine of this angle is found by the following calculation.
Figure 1.3.12(a) Angle inscribed in semicircle
Since its cosine is zero, the measure of ∠PRQ is π/2 so the angle is a right angle.
<< Previous Example Section 1.3
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document