4-1 Triangles HONORS GEOMETRY

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

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

Two Ways to Classify Triangles By Their Sides By Their Angles

Classifying Triangles By Their Sides Scalene Isosceles Equilateral

Scalene Triangles  

Isosceles Triangles  

Equilateral Triangle  

Classifying Triangles By Their Angles Acute Right Obtuse Equiangular

Contains ALL Acute Angles Acute Triangles Contains ALL Acute Angles      

Right Triangles Contains ONE Right Angle

Contains ONE Obtuse Angle Obtuse Triangles Contains ONE Obtuse Angle      

Equiangular Triangles        

Classify this triangle. Right Scalene

Classify this triangle.   Obtuse Isosceles

Classify this triangle.       Acute Scalene

Classify this triangle.   Acute Isosceles

Classify this triangle. Right Isosceles

How would you classify this triangle by sides?

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

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

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

How would you determine if it is a right triangle?

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

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.

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.

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.

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