HW CG –# 1 Intro to Coordinate Proofs
Name ________________________________
“Analytic Geometry” is the study of geometry using a coordinate system. The French mathematician Rene Descartes (1596-1650) is
traditionally considered the “father” of analytic geometry. The strength of analytic geometry lies in its use of tools from algebra to
solve geometry problems- here we will be connecting the ideas of the distance, slope and midpoint formulas to prove results about
geometric figures in the Cartesian Coordinate System.
Distance formula between two points, (𝑥1 , 𝑦1 )𝑎𝑛𝑑 (𝑥2 , 𝑦2 ) = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
Midpoint formula between two points, (𝑥1 , 𝑦1 )𝑎𝑛𝑑 (𝑥2 , 𝑦2 ) = (
𝑥1 +𝑥2 𝑦1 +𝑦2
2
,
2
)
Slope formula: 𝑚 =
𝑦2 −𝑦1
𝑥2 −𝑥1
y

Example A: Prove or disprove that the triangle with vertices A (4, 2),
B (-1, 4) , and C (2, -3) is an isosceles triangle.
a. Use the distance formula to find the length of each side:





x














b. Draw a conclusion based on our results. State whether or not the
triangle is isosceles and why.


C. What other conclusions can you make about the sides or angles of triangle ABC?
d. Suppose you do a translation according to the coordinate rule (x, y) -> (x-3, y+2.) Is triangle A’B’C’ still isosceles? Why or why
not?
y

Exercises

1.
Prove or disprove that the triangle with vertices R (-2, -2), S(1,
4) and T(4, -5) is equilateral. Include math calculations as well
as words in your proof.




x
















2.
Refer to the triangle from exercise 1 to prove or disprove the triangle with vertices R (-2, -2), S(1, 4) and T(4, -5) is a right
triangle. Include math calculations as well as words in your proof.
y

3.
Triangle ABC has vertices A(-4, 1), B (-3, 4) and C (-1, 1). Triangle
DEF has vertices D(2, -3) , E(5, -2) and F(2, 0). Prove or disprove
that the triangles are congruent. (Recall: triangles are congruent
iff you can show SAS, ASA, SSS, AAS or HL.) Include math
calculations as well as words in your proof.





x





















y
4.
Prove that the quadrilateral formed by points A (-3, 2,) B (1, -1),
C(1, 5), and D(5, 2) is a rhombus. Refer to specific properties of
rhombuses and support your proof with math calculations and
words.






x










