1. 4-1 Triangles
HONORS GEOMETRY

2. DEFINITION: A figure formed by three segments joining three non-collinear points.
Vertices: The three non-collinear points that form the figure. Points A, B & C. Sides: The three segments that join the vertices and form the triangle. Segments AB, BC & AC. Angles: The three angles formed by the sides of the triangle. Angles A, B & C. Named by: A symbol and three letters. Triangle ABC or ABC B C A

3. IMPORTANT TERMS:
Opposite: Segments AB, BC & AC are OPPOSITE Angles C, A & B respectively. Included: Angles A, B & C are the INCLUDED angles between Segments AC & AB, Segments BA & BC and Segments CA & CB respectively. B C A

4. Two Ways to Classify Triangles
By Their Sides By Their Angles

5. Classifying Triangles By Their Sides
Scalene
Isosceles
Equilateral

6. Scalene Triangles

7. Isosceles Triangles

8. Equilateral Triangle

9. Classifying Triangles By Their Angles
Acute
Right
Obtuse
Equiangular

10. Contains ALL Acute Angles
Acute Triangles
Contains ALL Acute Angles

11. Right Triangles
Contains ONE Right Angle

12. Contains ONE Obtuse Angle
Obtuse Triangles
Contains ONE Obtuse Angle

13. Equiangular Triangles

14. Classify this triangle.
Right Scalene

15. Classify this triangle.
Obtuse Isosceles

16. Classify this triangle.
Acute Scalene

17. Classify this triangle.
Acute Isosceles

18. Classify this triangle.
Right Isosceles

19. How would you classify this triangle by sides?

20. Review: The distance formula
To find the distance between two points in the coordinate plane…

21. Classify a triangle in a coordinate plane
EXAMPLE 1
Classify a triangle in a coordinate plane
Classify PQO by its sides.
STEP 1
Use the distance formula to find the side lengths. OP = y 2 – 1 ( ) x + = 2 – ( )
(– 1 ) + 5
2.2 OQ = y 2 – 1 ( ) x + 2 = – ( ) 6 + 3 45
6.7

22. Classify a triangle in a coordinate plane (continued)
EXAMPLE 1
Classify a triangle in a coordinate plane (continued) PQ = y 2 – 1 ( ) x + 3 – 2 ( ) 6 + =
(– 1 ) 50
7.1
PQO is a right scalene triangle.
ANSWER

23. How would you determine if it is a right triangle?

24. Classify a triangle in a coordinate plane (continued)
EXAMPLE 2
Classify a triangle in a coordinate plane (continued)
STEP 2
Check for right angles by checking the slopes.
There is a right angle in the triangle if any of the
slopes are perpendicular.
The slope of OP is
2 – 0
– 2 – 0 =
– 2.
The slope of OQ is
3 – 0
6 – 0 = 2 1 .
so OP OQ and POQ is a right angle.
Therefore, PQO is a right scalene triangle.
ANSWER

25. Example 3: Classify a triangle in the coordinate plane
Classify ΔABC by its sides. Then determine if the triangle is a right triangle. The vertices are A(0,0), B(3,3) and C(-3,3). STEP 1: Plot the points in the coordinate plane.

26. Example 3: (continued) Classify a triangle in the coordinate plane
STEP 2: Use the distance formula to find the side lengths: AB = BC = CA = Therefore, ΔABC is a ______________ triangle.

27. Example 3: (continued) Classify a triangle in the coordinate plane
STEP 3: Check for right angles by checking the slopes.
The slope of =
The slope of =
Therefore, ΔABC is a ______________ triangle.

28. HOMEWORK Pg. 240-245 #’s 4-37 all 44,46,49-52 all