1. Using Coordinate algebra, definitions, and properties
5D-3 Coordinate Proof
Using Coordinate algebra, definitions, and properties

2. Property of a Liquid
Takes the shape of the container while maintaining a constant volume

3. Liquid?

6. A Right Triangle is a triangle with one right angle.
What’s the definition,
theorem or property?
Prove that ∆𝐴𝐵𝐶 is a right triangle.
A Right Triangle is a triangle with one right angle.
Are there any right angles?
Or perpendicular sides?
What can you use to show this?
Strategy:
Find slopes of each side. Are they negative reciprocals?
If yes, the two sides form a right angle.
If there is one right angle, then it is a right triangle.

7. Proof: This is the format for a coordinate proof.
Prove is a right triangle.
Work Area: }
𝐴𝐵 𝑚= 𝑢𝑝 3 𝑜𝑣𝑒𝑟 4 = 3 4
𝐴𝐵 ⊥ 𝐵𝐶
𝐵𝐶 𝑚= 𝑑𝑜𝑤𝑛 4 𝑜𝑣𝑒𝑟 3 = −4 3
∠𝐵 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒
Statement:
Is a right triangle b/c is a right angle.

8. Prove ∆𝐴𝐵𝐶 is a right triangle.
Work Area: }
𝐶𝐵 𝑚= 𝑢𝑝 3 𝑜𝑣𝑒𝑟 4 = 3 4
𝐶𝐵 ⊥ 𝐶𝐴
𝐶𝐴 𝑚= 𝑑𝑜𝑤𝑛 4 𝑜𝑣𝑒𝑟 3 = −4 3
∠𝐶 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒
Statement:
Is a right triangle,
b/c
∆𝐴𝐵𝐶
∠𝐶 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒

9. Prove ∆𝐴𝐵𝐶 is an isosceles triangle.
Work Area: }
𝐶𝐵 𝑑= (5−1) 2 + (6−3) 2 = 5
𝐶𝐵 ≅ 𝐶𝐴
𝐶𝐴 𝑑= (4−1) 2 + (−1−3) 2 = 5
Statement:
Is an isosceles triangle,
b/c 𝐶𝐵 ≅ 𝐶𝐴
∆𝐴𝐵𝐶

10. Prove ABCD is a parallelogram.
Definition: Opposite sides are parallel.
Slopes
Property: Opposite sides are congruent.
Distance formula
You can choose either one, so
which seems easier to prove?

11. Prove ABCD is a parallelogram.
Work Area:
Statement:

12. Prove ABCD is a rectangle.
Definition: a Parallelogram
with one right angle.
Plan:
first, prove it’s a parallelogram
Second prove there is a right angle

13. Prove ABCD is a rectangle.
Work Area:
Statement: ABCD is a rectangle b/c
___________________ and
___________________

14. Definitions and Properties
Right triangle: a triangle with one right angle
Isosceles triangle: a triangle with two congruent sides
Parallelogram:
a) Quadrilateral with opposite sides parallel OR
b) Parallelogram Property: Opposite sides congruent
Rectangle: A parallelogram with one right angle

15. A Right Triangle is a triangle with one right angle.
What’s the definition,
theorem or property?
A Right Triangle is a triangle with one right angle.
Prove that ∆𝐴𝐵𝐶 is a right triangle.
Are there any right angles?
Or perpendicular sides?
What can you use to show this?
Strategy:
Find slopes of each side. Are they negative reciprocals?
If yes, the two sides form a right angle.
If there is one right angle, then it is a right triangle.

16. Proof: This is the format for a coordinate proof.
Prove is a right triangle.
Work Area:
_______ is a __________ __________
Statement:
Is a right triangle b/c _________________________

17. Prove ∆𝐴𝐵𝐶 is a right triangle.
Work Area:
_______ is a __________ __________
Statement:
Is a right triangle,
b/c
∆𝐴𝐵𝐶
∠𝐶 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒

18. Prove ∆𝐴𝐵𝐶 is an isosceles triangle.
Work Area:
Statement:
is an isosceles triangle, b/c ___________
∆𝐴𝐵𝐶

19. Prove ABCD is a parallelogram.
Definition: Opposite sides are parallel.
Slopes
Property: Opposite sides are congruent.
Distance formula
You can choose either one, so
which seems easier to prove?

20. Prove ABCD is a parallelogram.
Work Area:
Statement:

21. Prove ABCD is a rectangle.
Work Area:
Statement: ABCD is a rectangle b/c
___________________ and
___________________