Problem 43 - Algebra 2

Question

Mental Math Each given point is on the graph of the given equation. Use symmetry to find at least one more point on the graph. $$ (-2,0), 9 x^{2}+9 y^{2}-36=0 $$

Step-by-Step Solution

Answer

The virtual mirror image of the given point (-2,0) on the other side of the circle (across the y-axis) is (2,0).

Step 1: Identify the Circle's center and radius

Rearrange the given equation into the standard form of a circle's equation. This results in the equation \(x^2 + y^2 = 2^2\), which tells us that our circle's center is at the origin (0,0) and has a radius 2, because \(r^2 = 4\) so \(r = 2\)

Step 2: Understand the symmetry in a Circle

A circle is symmetrical about its center. In this case, our center is on the origin. When a point is on the x-axis, its symmetrical point will simply be directly across the circle on the same x-axis. In other words, its x-coordinate will be negated.

Step 3: Find the new symmetrical point

The given point is (-2,0). By applying the symmetry of the circle, simply negate the x-coordinate, resulting in the point (2,0) which is also on the circle.

Key Concepts

Symmetry in Geometry

Symmetry plays a crucial role in understanding the geometric properties of shapes, and it is especially straightforward when dealing with circles. A circle is defined by its center point and radius, and it exhibits a property called rotational symmetry. This means that a circle looks the same after any rotation around its center.

In particular, the circle mentioned in the exercise is symmetrical about its center, which lies at the origin (0,0). This tells us that for every point on the circle, there exists a diametrically opposite point across the circle's center.

• This symmetry simplifies many calculations since it allows us to predict locations of other points just by knowing one.
• For instance, for the given point (-2,0), you can find the symmetrical point (2,0) through this principle.

Understanding symmetry helps in visualizing and solving geometric problems more intuitively, making it easier to find related points or solve complex problems in a simpler way.

Center and Radius of a Circle

To identify the center and radius of a circle from its equation, we first need to rewrite the equation in the standard form of a circle, which is \[(x - h)^2 + (y - k)^2 = r^2\]Here, \(h, k\) signifies the circle's center, and \(r\) is the radius.

The exercise deals with the equation \[9x^2 + 9y^2 - 36 = 0\]Rearrange it to observe easily: dividing through by 9 gives the form \[x^2 + y^2 = 4\] telling us that:

• The center is at (0,0), as there is no \( h \) or \( k \) to subtract, leaving the x and y variables alone.
• The radius is 2, as indicated by \(r^2 = 4\), so \(r = \sqrt{4} = 2\).

Knowing these basic features, figure out quickly how the circle is positioned and how large it is. This knowledge is essential for graphing and further mathematical calculations.

Graphing Points on a Circle

Graphing points on a circle becomes much easier once you've established the circle's center and radius.

Given a point, you can check if it lies on the circle by plugging it back into the circle's equation. For example, if we graph the original point (-2,0) and check with \[x^2 + y^2 = 4\]we see that \[(-2)^2 + 0^2 = 4\]matches the right side, thus confirming the point is indeed on the circle.

• To use symmetry as the exercise suggests, find the symmetrical point by using the circle's properties.
• Often, if a point is \(a, b\), the symmetric point on a centered origin circle (like ours) can be derived by changing the sign of either coordinate depending on position and context.

With a compass and simple coordinates, you can visualize and accurately position further points on this circle. Knowing symmetry and having a keen eye for algebraic properties will aid in graphing quickly and accurately. This skill is quite beneficial in both educational settings and practical geometry applications.