Examples for

Common Core Math: High School Geometry: Expressing Geometric Properties with Equations

High school students use algebraic symbols and notation for the coordinate plane to describe geometric figures. In particular, students learn how to write equations to describe conic sections, which include circles and parabolas and, in advanced classes, ellipses and hyperbolas. Students also use algebraic notation with coordinates to prove geometric theorems and compute measurements such as perimeter and area.
Common Core Standards

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CCSS.Math.Content.HSG-GPE.B.7
gpe common core standard b5

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common core geometric properties high school standards
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Conic Sections

Define and describe circles, parabolas, ellipses and hyperbolas.

Identify circles with equations (CCSS.Math.Content.HSG-GPE.A.1):

equation of circle with center (2, 4) and radius 8
(x + 2)^2 + (y - 1)^2 = 2
circle centered at (3, 3) that contains (3, 7)
circle, diameter=10, center (0, 0)
x^2 + 14 x + y^2 - 8 y + 65 = 36 circle properties

Determine a parabola in terms of its focus and directrix (CCSS.Math.Content.HSG-GPE.A.2):

parabola with focus (2, 5) and directrix y = 1
focus (3, -2) directrix y = 4 parabola
focus and directrix of y = x^2 + 4x + 8
parabola with focus (3,4) and vertex (3,6)
directrix of parabola x^2+3y=16

Find equations for ellipses and hyperbolas (CCSS.Math.Content.HSG-GPE.A.3):

ellipse with foci at (-2, 5) and (9, 5) and major axis 16
(x^2)/5 + (y^2)/3 = 1
ellipse with equation (x-2)^2/8 + (y+1)^2/4 = 1
ellipse with axis lengths 2, 5 centered at (3,0)
hyperbola with center (–3, 2), vertices (–7, 2) and (1, 2), foci (–8, 2) and (2, 2)
((x - 2)^2)/4 - ((y - 2)^2)/8 = 1
hyperbola with center (100, 200) and focus (110, 180)
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Coordinate Geometry

Use coordinates to describe and reason with geometric figures.

Express geometric figures in terms of coordinates (CCSS.Math.Content.HSG-GPE.B.4):

rectangle {1, 1}, {1, 5}, {8, 5}, {8, 1}
{5, -3}, {6, 4}, {12, 6}, {7, 9}, {-1, 0} polygon
circle centered at (2, 2) and containing the point (2, 0)
circle through (0,0), (1,0), (0,1)

Find equations of parallel and perpendicular lines (CCSS.Math.Content.HSG-GPE.B.5):

line through (1, 4) that is parallel to y = -2x + 7
line parallel to y = 2x + 127 through (42, 248)
line through (5, 5) and perpendicular to -(1/4)x - 9
line perpendicular to 2x + 2 through (2, -3)

Partition a line segment (CCSS.Math.Content.HSG-GPE.B.6):

midpoint of line between (2, 5) and (-1, 8)
(36, 102) and (50, 110) midpoint

Compute areas and perimeters of polygons (CCSS.Math.Content.HSG-GPE.B.7):

area of rectangle {-2, 4}, {-2, -5}, {6, -5}, {6, 4}
triangle {{4, 5}, {8, 8}, {7, -1}} perimeter
polygon {{0, 1}, {4, 1}, {6, 4}, {4, 8}, {0, 8}} area
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