Read and reconstruct an equation
For (x−2)²+(y+3)²=25, the center is (2,−3) and the radius is five. In expanded form x²+y²−4x+6y−12=0, group the coordinate terms and add four and nine to complete both squares. The right side becomes 12+4+9=25. Adding a square-completing term on only one side changes the circle.
Arc length depends on the angle unit
With radius six and central angle 2/3 radians, arc length is rθ=4. If the angle is instead 60 degrees, use 60/360 of the circumference to obtain 2π. Radians encode the ratio of arc length to radius, so the radian formula does not include a degree conversion factor.
A sector is an area
A 90-degree sector of radius four has area (90/360)π(4²)=4π. Its arc length is 2π, a different quantity with different units. The same fraction of a full turn applies to each whole-circle measure, but the whole measure must match the requested dimension.
Transfer and check
Complete squares in x²+y²−6x−8y=0. The result (x−3)²+(y−4)²=25 has radius five. Check that the origin lies on this circle, since its distance from the center is five. Explain why the constant zero in expanded form did not mean radius zero.
A circle has radius 4 cm. A central angle measures 2/3 radians. What is the corresponding arc length in centimeters?
- For a radian measure, arc length is s=rθ.
- Multiply 4 by 2/3.
- The arc length is 8/3 cm; radians relate arc length to radius.
Compare the other answer choices
Divide by the radian measure instead of multiplying.
Insert an unnecessary factor of one half.
Return the angle rather than a length.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
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