Warm Up for Section 1.1 Simplify: (1). (2). Use the triangle below to answer #3, #4: (3). Find x. (4). If a = 5, b = 3, find c. 40 o a b c xoxo

Answers for Warm Up, Section 1.1 Simplify: (1). (2). Use the triangle below to answer #3, #4: (3). Find x. (4). If a = 5, b = 3, find c. 40 o a b c xoxo

Work for Answers to WU, Section 1.1 (1). (2). (3). x = 90 – 40 (4). a 2 + b 2 = c 2 = = c = c 2 34 = c 2 = c

Special Right Triangles Section 1.1 Essential Question: What is the relationship between the lengths of the legs of a 45°–45°–90° triangle and a 30°–60°–90° triangle? Standard: MM2G1a, b

Vocabulary Right Triangle: A triangle containing one angle that measures exactly 90 degrees. Hypotenuse: The longest side of a right triangle. Reference angle: The measured, or known angle in a right triangle other than the 90° angle.

Investigation 1: With your partner, complete each step in the investigation then answer questions Step 1: Using the grid paper provided and a straightedge, draw a square with side length 5 cm. Step 2: Label the vertices of the square A, B, C, and D. Label each side with its length. Step 3: Using a straightedge, draw diagonal.

Investigation 1: A B C D 5 cm

Answer the following questions: (1). m  D = ____ o (2). m  ACD = ____ o (3). m  DAC = ____ o (4). DC = ____ (5). AD = ____ (6).  ADC is (acute, right, obtuse). (7).  ADC is (isosceles, scalene, equilateral). (8). Using the Pythagorean Theorem, find AC. Be sure to write your answer in simple radical form cm

45° 5 5 a 2 + b 2 = c = x = x 2 50 = x 2 x

Look at two additional 45 o -45 o -90 o triangles and determine the length of the hypotenuse, x. Be sure to write your answer in simple radical form.

45° 3 3 a 2 + b 2 = c = x = x 2 18 = x 2 x Question 9: Find x

45° 8 8 a 2 + b 2 = c = x = x = x 2 x Question 10: Find x

45° x x (a). Length of hypotenuse = length of leg times. (b). Length of legs = length of hypotenuse divided by. Summary: In a 45 o -45 o -90 o triangle

Investigation 2: With your partner, complete the following regarding equilateral  ABC where AB =10: Step 1: Label the length of each edge. Step 2: Label the measure of  B and  C. Step 3: Using a straightedge, draw and label altitude. Step 4: Label the length of and. Step 5: Label the measure of  BAD and  CAD. Step 6: Label the measure of  ADC. Step 7: Using the Pythagorean Theorem, find AD.

° 30° 60° a 2 + b 2 = c x 2 = x 2 = = x 2 30° A B C D x

Investigation 2: Note: the two legs of a 30 o -60 o -90 o triangle are NOT equal in measure. The longer leg will always be opposite the ___ o angle. The shorter leg will always be opposite the ___ o angle

Consider the 30 o -60 o -90 o right triangle created from an equilateral triangle pictured at right. (13). The long leg is segment ______ and the short leg is segment _______. (14). Use the Pythagorean Theorem to find RT. RT ST ° 30° R TS

° 30° a 2 + b 2 = c x 2 = x 2 = = x 2 R TS

2x2x x 60° 30° Length of hypotenuse = length of short leg times 2 Length of long leg: length of short leg times Length of short leg: half the length of hypotenuse or the length of the long leg divided by Summary: In a 30 o -60 o -90 o triangle:

Check for Understanding: Find the missing edge lengths for each triangle: Example 13:

Check for Understanding: Find the missing edge lengths for each triangle: Example 14: 60 o 30 o

Check for Understanding: Find the missing edge lengths for each triangle: Example 15:

Check for Understanding: Find the missing edge lengths for each triangle: Example 14: 60 o 30 o

Check for Understanding: Find the missing edge lengths for each triangle: Example 14: 60 o 30 o

Check for Understanding: Find the missing edge lengths for each triangle: Example 14: 60 o 30 o

Check for Understanding: Find the missing edge lengths for each triangle: Example 14: 60 o 30 o