Practice 6.2.1: Using Coordinates to Prove Geometric Theorems
About Circles and Parabolas
Use the given information to prove or disprove each statement. Justify your reasoning.


1. Prove or disprove that point Q 1,  2 lies on the circle centered at the
origin R and passing through the point A (0, –3).
2. Prove or disprove that point A (0, 7) is on the circle centered at the origin R and
passing through the point P 5, 2 6 .


3. Given the points P (–2, 2), Q (4, 8), and R (0, 0), prove or disprove that the
1
 1 
points are on the parabola with focus F 0,  and directrix y   .
 2  
2
4. Given the points A (–2, –8), B (1, –2), and C (2.25, –10.125), prove or disprove
  1  and
that the points are on the quadratic function graph with focus F 0,

8 
1
directrix y  .
8
5. Prove or disprove that the points A (5, 1), B (2, –2), C (6, –2), and D (1, –7)
are all on the quadratic function graph with vertex V (4, 2) that passes through
E (0, –14).
continued
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Lesson 2: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas
PRACTICE
UNIT 6 • MODELING GEOMETRY
Lesson 2: Using Coordinates to Prove Geometric Theorems About
Circles and Parabolas
6. Prove or disprove that the point A (5, 6) lies on the parabola with focus F (5, 1)
and directrix x = –1.
7. Prove or disprove that the points A (–5, 12), B (5, 12), and C (0, –13) are the
vertices of an isosceles triangle inscribed in the circle centered at the origin Q


and passing through the point P 10, 69 .
8. The diagram below shows a target at a carnival dart game. The diagram is on
a coordinate system. A player wins a prize by hitting the shaded ring. The ring
is formed by two circles. Both circles have center C (12, 12). One circle passes
through P (6, 20) and the other circle passes through P (12, 23). Natasha throws
1
2
a dart and hits the point Q (19, 4). Does she get a prize? Justify your answer.
y
C (12, 12)
x
continued
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Lesson 2: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas
PRACTICE
UNIT 6 • MODELING GEOMETRY
Lesson 2: Using Coordinates to Prove Geometric Theorems About
Circles and Parabolas
9. An art student created the following graph to represent the letter M. To create
the image, she graphed two parabolas intersecting at the point A. The parabolas
are described as follows:
•
a parabola with vertex V 1 (6, 8) and focus F 1 (6, 5)
•
a parabola with vertex V 2 (18, 8) and focus F 2 (18, 5)
Prove or disprove that point A has coordinates (12, 5).
y
A
x
10. The diagram below represents a suspension bridge. The curve is a portion of
a parabola. The parabola has vertex V (0, 10) and passes through the point
(20, 10.4). Prove or disprove that all points on the parabola are equidistant
from the point (0, 260) and the line y = –240.
U6-64
Lesson 3: Solving Systems of Linear Equations and Circles
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Lesson 3: Solving Systems of Linear Equations and Circles