1. EXAMPLE 1 Classify triangles by sides and by angles SOLUTION The triangle has a pair of congruent sides, so it is isosceles. By measuring, the angles are 55°, 55°, and 70°. It is an acute isosceles triangle. Support Beams Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles.

2. EXAMPLE 2 Classify a triangle in a coordinate plane SOLUTION STEP 1 Use the distance formula to find the side lengths. Classify PQO by its sides. Then determine if the triangle is a right triangle. OP= y 2 –y 1 ( ) 2 x 2 –x 1 ( ) 2 + = 2–0 ( ) 2 (– 1 ) 0 ( ) 2 + – = 5 2.2 OQ= y 2 –y 1 ( ) 2 x 2 –x 1 ( ) 2 + 2 = –0 ( )6 0 ( ) 2 + – 3 = 45 6.7

3. EXAMPLE 2 Classify a triangle in a coordinate plane PQ= y 2 –y 1 ( ) 2 x 2 –x 1 ( ) 2 + 3– 2( ) 2 6 ( ) 2 + – = (– 1 ) = 50 7.1 STEP 2 Check for right angles. The slope of OP is 2 – 0 – 2 – 0 = – 2. The slope of OQ is 3 – 0 6 – 0 = 2 1. 1 The product of the slopes is – 2 2 = – 1, so OP OQ and POQ is a right angle. Therefore, PQO is a right scalene triangle. ANSWER

4. GUIDED PRACTICE for Examples 1 and 2 1. Draw an obtuse isosceles triangle and an acute scalene triangle. obtuse isosceles triangle B AC acute scalene triangle P Q R

5. GUIDED PRACTICE for Examples 1 and 2 2. Triangle ABC has the vertices A(0, 0), B(3, 3), and C(–3, 3). Classify it by its sides. Then determine if it is a right triangle. SOLUTION STEP 1 Use the distance formula to find the side lengths. AB= y 2 –y 1 ( ) 2 x 2 –x 1 ( ) 2 + = 3–0 ( ) 2 ( 3 ) 0 ( ) 2 + – BC= y 2 –y 1 ( ) 2 x 2 x 1 ( ) 2 + 2 = – 3 ( )–3–3 3 ( ) 2 + – 3 = 18 4.2 = 400 20

6. GUIDED PRACTICE for Examples 1 and 2 AC= y 2 –y 1 ( ) 2 x 2 –x 1 ( ) 2 + = 3– 0( ) 2 0 )(–3(–3 ( ) 2 + – = 18 4.2 STEP 2 Check for right angles. The slope of AB is 3 – 0 = 1.1. The product of the slopes is 1(– 1) = – 1, so AB AC and BAC is a right angle. The slope of AC is 3 – 0 – 3 – 0 =. – 1 Therefore, ABC is a right Isosceles triangle. ANSWER