Problem 73 - Algebra 2

Question

Circle \(A\) has equation \((x+5)^{2}+y^{2}=169 .\) The diameter of circle \(B\) is one fourth as long as the diameter of circle \(A\) . What is the radius of circle \(B ?\)

Step-by-Step Solution

Answer

The radius of circle B is 3.25.

Step 1: Identify the radius of circle A

The equation of circle A is given by \((x+5)^{2}+y^{2}=169\). In the standard form of the equation of a circle, the right side of the equation gives \(r^2\). So, the radius \(r_A\) of circle A can be found by taking the square root of 169, which equals 13.

Step 2: Calculate the diameter of circle A

The diameter of a circle is twice its radius. Therefore, the diameter \(d_A\) of circle A is \(2 * r_A = 2 * 13 = 26\).

Step 3: Determine the diameter of circle B

It's given that the diameter of circle B is one fourth the diameter of circle A. So, the diameter \(d_B\) of circle B is \(1/4 * d_A = 1/4 * 26 = 6.5\).

Step 4: Find the radius of circle B

Finally, the radius \(r_B\) of circle B is half its diameter. Therefore, \(r_B = d_B / 2 = 6.5 / 2 = 3.25\)

Key Concepts

Circle equation

Understanding the equation of a circle is essential in geometry. The general form of a circle's equation is \((x-h)^{2} + (y-k)^{2} = r^{2}\), where:\

\
• \(h\) and \(k\) are the coordinates of the circle’s center.\
\
• \(r\) is the radius of the circle.\
\

\For example, the circle equation \((x+5)^{2} + y^{2} = 169\) can be rewritten as \((x-(-5))^{2} + (y-0)^{2} = 13^{2}\), showing:\

\
• The circle is centered at \((-5, 0)\).\
\
• The radius \(r\) is 13, as \(169\) is the square of \(13\).\
\

\This understanding allows circles to be represented graphically and analyzed mathematically, providing insights into their geometric properties.

Radius calculation

Calculating the radius of a circle from its equation is straightforward. The key lies in recognizing the \(r^{2}\) component in the equation, which directly connects to the circle's radius. For instance, in \((x+5)^{2} + y^{2} = 169\):\

\
• The right-hand side \(169\) equals \(r^{2}\).\
\
• Taking the square root of \(169\) gives \(r = 13\).\
\

\With this radius in hand, various calculations, like finding circumference or area, become simple. Always make sure to extract root values carefully, ensuring the correct interpretation of the circle’s size.

Diameter of a circle

The diameter of a circle is twice the length of its radius. It runs through the center, touching two opposite points on the circle's boundary. This relationship is useful for many applications, as shown in the context of the problem with circles A and B. For circle A with radius \(13\):\

\
• The diameter \(d_A\) is \(2 \times 13 = 26\).\
\
• For circle B, with its diameter defined relative to circle A, \(d_B = 1/4 \times 26 = 6.5\).\
\

\Knowing the diameter helps in visualizing and comparing different circles, as well as solving more complex geometry problems involving circles.

Mathematical problem solving

Solving mathematical problems involves a structured approach. With circles, begin by identifying the components from the equation. In the given problem with circles A and B, steps include: \
\

\
• Finding circle A's radius from \((x+5)^{2} + y^{2} = 169\).\
\
• Calculating diameter \(d_A = 2 \times 13 = 26\).\
\
• Determining circle B’s diameter as a fraction of \(d_A\).\
\
• Dividing circle B's diameter by 2 to find \(r_B = 3.25\).\
\

\This step-by-step problem-solving process ensures clarity in mathematical reasoning, breaking down complex problems into manageable parts.