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
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Concept explainers
Cylinders
A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.
Cones
A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is drawn by joining the apex to all points on the base, using segments, lines, or half-lines, provided that the apex and the base both are in different planes.
Question
Transcribed Image Text:**Geometry Problem: Finding Segment Lengths in a Circle**
**Problem Statement:**
XZ is a diameter of the circle shown. The radius of the circle is 13 feet and YZ = 24 feet. Find XY.
**Explanation:**
In this problem, we are given a circle with a diameter XZ and a point Y on the circumference of the circle. Important details given include:
1. The radius of the circle is 13 feet.
2. The length of segment YZ is 24 feet.
We need to determine the length of segment XY.
**Steps to Solve:**
1. **Understand the Given Information:**
- The radius of the circle is 13 feet.
- The diameter is twice the radius, so XZ = 26 feet.
- YZ = 24 feet.
2. **Diagram Explanation:**
- The circle has its diameter XZ.
- Point Y is somewhere on the circumference, making the segments XY, YZ, and XZ form a triangle within the circle.
3. **Using Geometry Principles:**
- Since the problem involves a circle and its diameters, we can apply the Pythagorean theorem in the circle.
- The triangle XYZ with XZ as the hypotenuse (26 feet) and YZ (24 feet) can be solved for XY.
4. **Application of Pythagorean Theorem:**
- \(XZ^2 = XY^2 + YZ^2\)
- \(26^2 = XY^2 + 24^2\)
- \(676 = XY^2 + 576\)
- Solving for XY, subtract 576 from both sides:
\(676 - 576 = XY^2\)
\(100 = XY^2\)
\(XY = \sqrt{100}\)
\(XY = 10\) feet
Therefore, the length of segment XY is 10 feet.
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