Special Cases, for Right Triangles 60° 30° 45° 45° 45° 90° Triangles 45°

30° 60° 90° triangles have just been born 1. An equilateral triangle is also equiangular, all angles are the same. 60° 2 2. Let’s draw an Altitude from one of the vertices. Which is also a Median and Angle bisector. 1 30° 2 60° 30° 3. The bisected side is divided into two equal segments and the bisected angle has now two 30° equal angles. 2 60° Congratulations! Two 30° 60° 90° triangles have just been born Oooh …and you watched!!!!

And the other leg is times as big, 1 * = Let’s separate the top triangle and label the unknown side as z. 1 60° 30° 2 60° 30° 2 60° apply the Pythagorean Theorem to find the unknown side. 1 1 2 2 = z + 1 2 4 = z + 1 2 -1 -1 z = 3 z 3 = z 2 3 = z 2 When, the smallest side is equal to 1, the hypotenuse is 2 times as big, 1 * 2 = 2 And the other leg is times as big, 1 * = Can we generalize this result for all 30°-60°-90° right triangles? z = 3

3 3 3 60° 30° 1 2 60° (2) 4 2 (2) 2 1 Yes it works! 30° (2) 1 2 (.5) Is this true for a triangle that is twice as big? Is this true for a triangle that is half the original size? Yes , it still works. If we know 1 side length of a 30-60-90 triangle, we can use this pattern to find the other 2 sides

the hypotenuse is twice as long as the shorter leg, and 3 s 60° 2 30° In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg. 3

Find the values of the variables Find the values of the variables. Round your answers to the nearest hundredth. y 2x = 14 30° 2x = 14 x 2 2 14 60° x =2 Is this 30°-60°-90°? 90°-30°=60° y = x 3 Then we know that: y = 2 3 2 3 s 60° 30°

. 90 = x 3 2x = y 90 = x 3 ( ) = y 2 3 3 y 90 = y 3 90 3 = x 3 y= 3 OR Find the values of the variables. Round your answers to the nearest unit. 90 = x 3 2x = y 90 = x 3 ( ) = y 2 3 30 30° 3 y 90 = y 3 60 . 90 3 = x 3 y= 3 60 OR 60° 3 90 = x x 3 ( ) 2 y =104 Is this a 30°-60°-90°? 90°-60°=30° 3 90 = x 2 3 s 60° 30° 3 30 = x 3 30 x= x = 52 OR

Find the values of the variables. Find the exact answer. 3 2x = y 60° 30 = x 3 ( ) = y 2 3 10 3 = y 3 20 y . 30 3 = x 3 30 y= 3 20 30° 3 30 = x 3 ( ) 2 Is this a 30°-60°-90°? 90°-60°=30° 3 30 = x 2 3 s 60° 30° 3 10 = x 3 10 x=

What kind of right triangles are formed? 1 Let’s draw a diagonal for the square. The diagonal bisects the right angles of the square. What kind of right triangles are formed?

1 45° 1 45° 45° y y = 1 + 1 2 The triangles are 45°-45°-90° y = 1 + 1 2 Let’s draw the bottom triangle and label the hypotenuse as y y = 2 2 Let’s apply the Pythagorean Theorem to find the hypotenuse. y = 2 2 y = 2

45-45-90 Right Triangle x x Can we generalize our findings?

In a 45°-45°-90° triangle, the hypotenuse is times as long as a leg And both legs are the same size. 2 45° s 2 s 45° s

. 36 = x 2 36 = x 2 36 2 x If y = x 36 2 = x 2 then 2 y = OR 2 = x y 2 Find the values of the variables. Round your answers to the nearest tenth. 36 = x 2 36 = x 2 45° 36 2 x . If y = x 36 2 = x 2 then 2 18 y = 45° OR 2 36 = x y 2 ( ) y = 25.5 Is this a 45°-45°-90°? 90°-45°=45° s 2 45° 2 36 = x 2 18 = x 2 18 x = OR x = 25.5

. 42 = x 2 42 = x 2 42 2 x If y = x 42 2 = x 2 then 2 y = 2 = x y 2 Find the values of the variables. Give an exact answer. 42 = x 2 42 = x 2 45° 42 2 x . If y = x 42 2 = x 2 then 2 21 y = 45° 2 42 = x y 2 ( ) Is this a 45°-45°-90°? 90°-45°=45° s 2 45° 2 42 = x 2 21 = x 2 21 x =

Find the values of the variables. Give the exact answer. 45° 2 x y= y x 2 21 y = 45° 21 Is this a 45°-45°-90°? 90°-45°=45° s 2 45°