Problem 53 - Algebra 2

Question

Which equation does NOT represent a line of symmetry for the circle with equation \(x^{2}+y^{2}=100 ?\) \(\begin{array}{llll}{\text { F. } x=0} & {\text { G. } y=\frac{1}{2} x} & {\text { H. } y=x} & {\text { J. } y=x+1}\end{array}\)

Step-by-Step Solution

Answer

The equation that does NOT represent a line of symmetry for the circle is \( y= x + 1 \).

Step 1: Check Line F

The line \( x=0 \) clearly passes through the point (0,0) as when \( x=0 \), \( y \) can be any real number.

Step 2: Check Line G

The line \( y=\frac{1}{2}x \) also passes through the point (0,0) because when \( x=0 \), \( y \) also equals 0.

Step 3: Check Line H

The line \( y=x \) too passes through the point (0,0) since when \( x=0 \), \( y = 0 \).

Step 4: Check Line J

Line \( y= x + 1 \) does not pass through the point (0,0) as when \( x=0 \), \( y \) equals 1, not zero.

Key Concepts

Equations of Symmetry

Symmetry in geometry refers to the balanced and proportional arrangement of points or shapes around a central line or point. When it comes to circles, an equation of symmetry is typically a line that divides the circle into mirror-image halves.
For the circle described by the equation \(x^2 + y^2 = 100\), we're determining which lines represent symmetry axes.

• The line \(x = 0\) is a vertical line through the center of the circle, which makes it a line of symmetry.
• The line \(y = \frac{1}{2}x\) also intersects the circle at its center, maintaining symmetry.
• The line \(y = x\) follows a similar logic, dividing the circle equally through its center.
• However, \(y = x + 1\) doesn’t pass through the circle’s center. Thus, it is not a line of symmetry for this circle.

Remember, a line of symmetry for a circle must pass through its center.

Geometry

Geometry is the branch of mathematics dealing with shapes, sizes, and the properties of space. It helps us understand and articulate the characteristics of various geometric figures.
With circles, geometry examines how they interact with lines, planes, and space. The circle \(x^2 + y^2 = 100\) represents a perfect geometric shape.
In geometric terms, a circle is the set of all points equidistant from a fixed point, known as the center.
Understanding basic geometric principles allows us to determine symmetry, analyze shapes, and solve problems involving shapes like this circle.

Circles

A circle is a simple closed shape. All points on a circle are the same distance from its center. In our exercise, the circle is defined by \(x^2 + y^2 = 100\), which suggests that every point is 10 units away from the center (0, 0), as the square root of 100 is 10.
Circles can have many lines of symmetry, but they must intersect at the circle's center.

• All diameter lines are axes of symmetry for a circle.
• These axes split the circle into identical parts.
• Non-diameter lines, such as \(y = x + 1\), cannot serve as symmetry lines because they don’t pass through the center.

Knowing the basics of circles helps us to recognize symmetry lines correctly.

Coordinate Geometry

Coordinate geometry, or analytic geometry, connects algebra and geometry using coordinates to define shapes and positions in space. In this exercise, it allows us to visualize and verify lines as symmetry lines by placing them on a graph with the circle.
The circle \(x^2 + y^2 = 100\) is centered at the origin (0,0), with a radius of 10.

• Lines \(x=0\) and \(y=x\) cutting through (0,0) clearly appear as lines of symmetry.
• The line \(y=\frac{1}{2}x\) also goes through the origin.
• But \(y=x+1\) is parallel to \(y=x\) but offset by 1 unit on the y-axis, hence not a symmetry line.

By using coordinate geometry, we can accurately determine which lines show symmetry for this circle.