Special Right Triangles. Draw 5 squares with each side length increasing by 1 1 2 3 4 5.

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Presentation transcript:

Special Right Triangles

Draw 5 squares with each side length increasing by

Find the length of each diagonal

1 a 2 + b 2 = c = c = c 2 2 = c 2  2 = c

Find the length of each diagonal 1 2 a 2 + b 2 = c = c = c 2 8 = c 2  8 = c 2  2 = c

Find the length of each diagonal a 2 + b 2 = c = c = c 2 18 = c 2  18 = c 3  2 = c

Find the length of each diagonal a 2 + b 2 = c = c = c 2 32 = c 2  32 = c 4  2 = c

Find the length of each diagonal a 2 + b 2 = c = c = c 2 50 = c 2  50 = c 5  2 = c

Do you see the pattern  2 4  2 3  2 2  2  2 Side length time square root of 2

Now, what do you know about squares and the triangle formed by the diagonal. A square has 4 equal sides and 90° angles The triangle is isosceles right

If the triangle is isosceles right, then what do you know about the base angles? 45°

An isosceles right triangle is a triangle 45°

triangles follow the same pattern we found in the squares. 45° a a  2 So if you know one side you can find the other.

Find the missing sides 45° 8 8  2 c 8

Find the missing sides 45° a 7  2 7 7

Find the missing sides  2 c 25

Find the missing sides a 16  2 16

triangles will have two basic diagrams that you will need to recognize. 45°

You will have to know triangles… the hypotenuse = leg  2 45°

OR.. the leg = hypotenuse  2 45°

There is a second “special right triangle”… Obviously if one angle is 30° then the other is 60° 30° Angle Sum of Triangle

triangle The short leg (a) is always opposite the 30° angle 30° a

triangle The hypotenuse is always 2 times short leg 30° 2a a

Draw five triangles Each short leg increasing by 1 30° ° Find the long leg b b b b b

a 2 + b 2 = c b 2 = b 2 = 4 b 2 = 4-1 b 2 = 3 b =  3 30° 2 1 b

Find the long leg a 2 + b 2 = c b 2 = b 2 = 16 b 2 = 16-4 b 2 = 12 b =  12 b = 2  3 30° b b

Find the long leg a 2 + b 2 = c b 2 = b 2 = 36 b 2 = 36-9 b 2 = 27 b =  27 b = 3  3 30° b b 6 3 b

Find the long leg a 2 + b 2 = c b 2 = b 2 = 64 b 2 = b 2 = 48 b =  48 b = 4  3 30° b b 6 3 b 8 4 b

Find the long leg 30° b b 6 3 b 8 4 b 10 5 b a 2 + b 2 = c b 2 = b 2 = 100 b 2 = b =  75 b = 5  3

Do you see the pattern? 30°  3 30° 4 4  3 30° 5 5  3 2  3  3 Long leg = short leg time the square root of 3

triangle 30° 2a a a3a3

Find the missing side 30° h 8 8383 16 b