Problem Analysis:
The task involves analyzing the given information about a circle and its properties, such as the center, radius, diameter, and equation. The goal is to determine whether each statement is
Always True (AT),
Sometimes True (ST), or
Never True (NT) based on the provided context.
#### Given Information:
1.
Center of the Circle: \((h, k)\)
2.
Radius of the Circle: \(r\)
3.
Diameter of the Circle: \(d = 2r\)
4.
Equation of the Circle: \((x - h)^2 + (y - k)^2 = r^2\)
We need to evaluate each statement in light of this information.
---
Statement 1: "The center of the circle is always at the origin."
-
Analysis:
- The center of the circle is given as \((h, k)\).
- For the center to be at the origin, both \(h\) and \(k\) must be zero (\(h = 0\) and \(k = 0\)).
- However, the problem does not specify that \(h\) and \(k\) are zero; they can take any real values.
- Therefore, the center is not always at the origin.
-
Conclusion:
Sometimes True (ST)
---
Statement 2: "The length of the diameter is always twice the radius."
-
Analysis:
- The diameter \(d\) is defined as \(d = 2r\).
- This relationship is a fundamental property of circles and holds true for all circles, regardless of their size or position.
- There are no exceptions to this rule.
-
Conclusion:
Always True (AT)
---
Statement 3: "The equation of the circle can be written without knowing the radius."
-
Analysis:
- The standard form of the equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\).
- To write this equation, we need to know the values of \(h\), \(k\), and \(r\). Without knowing \(r\), the equation cannot be fully specified.
- For example, if we only know the center \((h, k)\), we cannot write the complete equation because \(r^2\) is missing.
-
Conclusion:
Never True (NT)
---
Statement 4: "If you move the circle 3 units to the right, the new center will have coordinates \((h + 3, k)\)."
-
Analysis:
- Moving a circle horizontally affects only the \(x\)-coordinate of its center.
- If the original center is \((h, k)\), moving it 3 units to the right means adding 3 to the \(x\)-coordinate.
- Thus, the new center becomes \((h + 3, k)\).
-
Conclusion:
Always True (AT)
---
Final Answer:
Based on the analysis above, the answers are:
1.
Sometimes True (ST)
2.
Always True (AT)
3.
Never True (NT)
4.
Always True (AT)
$$
\boxed{\text{ST, AT, NT, AT}}
$$
Parent Tip: Review the logic above to help your child master the concept of independent variable vs dependent variable worksheet.
