Geometry Items to Support Formative Assessment
Unit 3: Circles, Proof, and Construction
Translate between the geometric description and the equation for a conic section.
G.GPE.A.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem;
complete the square to find the center and radius of a circle given by an equation.
Task:
A landscape company has been contracted to build a circular garden with diameter 20 feet. They plan to
place a fountain at its center. The yard has been graphed with vertices (0,0), (19,0), (19,11), and (0, 11).
The house’s coordinates are (3,1), (3, 13), (13, 7), and (3, 7). One unit on the graph equals five feet.
Write an equation for the border of the garden, and state the coordinates for the fountain. The garden’s
border may not touch the house.
Possible Solutions: (x - 16)2 + (y - 9)2 = 102 with fountain coordinates of (16, 9) or (x - 16)2 + (y - 3)2 =
102 with fountain coordinates of (16, 3)
Item 1:
Write the equation for the circle shown below:
Radius = 5, Center = (6,4)
(x-6)2 + (y - 4)2 = 52
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Item 2:
Complete the square to find the center and radius of a circle given by the equation
x2 + 6x + y2 - 14y + 42 = 0
Solution: (x + 3)2 + (y - 7)2 = 42, so the radius = 4, and the center is (-3, 7)
Item 3:
1. Find the equation of circle A.
2. Find the points of intersection of the circle and the x-axis (round to the nearest hundredth)
Solution:
1. (x – 3)2 + (y – 1)2 = 9
2. Substitute 0 for y: (0.17, 0) or (5.83, 0)
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. i.e. prove or disprove that
the point lies on the circle centered at the origin and containing the point (0, 2).
Task:
A farmer tethers her goat to a stake with a rope 15 feet long. Everyday she must move the stake so the
goat can eat fresh grass. She has a small herb garden whose vertices are (2, 1), (6, 1), (6, 6), and (2, 6). If
she moves the stake to the points listed below, state whether the goat can nibble on the herb garden.
Stake coordinates:
1. (20, 12)
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2. (18, 12)
3. (16, 1)
Solutions:
(20, 12) no
(18, 12) yes
(16, 1) yes
Item 1:
Show the point (11,43) lies on the circle that is centered at (2,3) and has a radius of 41.
(x-h)2 + (y-k)2 = r2
Solution: (11-2)2 + (43-3)2 = 412
Item 2:
Are the points (4, 4), (0, -3), and (5,2) on, inside, or outside the circle whose diameter is defined by (-3,2)
and (5, -6)?
Solution:
Center is at (1, -2), Radius = 4 2
Equation of circle: (x-1)2 + (y+2)2 = 32
For (4,4): (4-1)2 + (3+2)2 > 32 (outside)
For (0, -3): (0-1)2 + (-3+2)2 < 32 (inside)
For (5, 2): (5-1)2 + (2+2)2 = 32 (on)
Item 3:
A circle centered at the origin with a radius of 13 passes through points with integer coordinates 12 times.
How many of those points can you find?
Solution:
13 is the hypotenuse of the Pythagorean triple 5-12-13. It will pass through points (13, 0), (5, 12), (12, 5).
These points can be reflected around the circle to yield the other 9 points.
If students have found a few points, but can’t figure out the rest, suggest transformations (rotation or
reflection) and the nature of circles (all points equidistant) as a strategy to find others.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.