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Obj. 24 Special Right Triangles The student is able to (I can): Identify when a triangle is a 45-45-90 or 30-60-90 triangle Use special right triangle relationships to solve problems
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Obj. 24 Special Right Triangles

Jul 23, 2015

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  • Obj. 24 Special Right Triangles

    The student is able to (I can):

    Identify when a triangle is a 45-45-90 or 30-60-90 triangle

    Use special right triangle relationships to solve problems

  • Consider the following triangle:

    To find x, we would use a2 + b2 = c2, which gives us:

    What would x be if each leg was 2?

    1

    1 x

    2 2 2

    2

    1 1 x

    x 1 1 2

    x 2

    + =

    = + =

    =

  • Again, we will use the Pythagorean Theorem

    Simplifying the radical, we can factor to give us

    Do you notice a pattern?

    2

    2 x

    2 2 2

    2

    2 2 x

    x 4 4 8

    x 8

    + =

    = + =

    =

    82 2.

  • Thm 5-8-1 45-45-90 Triangle Theorem

    In a 45-45-90 triangle, both legs are congruent, and the length of the hypotenuse is times the length of the leg.

    2

    x

    x x 245

    45

  • Example Find the value of x. Give your answer in simplest radical form.

    1.

    2.

    3.

    45

    x

    8

    8 2

    x7

    7 2

    9 2x

    9 29

    2=

  • If we know the hypotenuse and need to find the leg of a 45-45-90 triangle, we have to divide by . This means we will have to rationalize the denominator, which means to multiply the top and bottom by the radical.

    The shortcut for this is to divide the hypotenuse by 2 and then multiply by

    2

    16 x

    16 16 2x

    2 2 2

    = =

    16 28 2

    2= =

    2.

    16x 2 8 2

    2= =

  • Examples Find the value of x.

    1.

    2.

    x

    45 20

    20x 2 10 2

    2= =

    x

    5

    5x 2

    2=

  • Thm 5-8-2 30-60-90 Triangle Theorem

    In a 30-60-90 triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is times the length of the shorter leg.

    Note: the shorter leg is always opposite the 30 angle; the longer leg is always opposite the 60 angle.

    3

    x

    2xx 3

    60

    30

  • Examples Find the value of x. Simplify radicals.

    1. 2.

    3. 4.

    7

    x

    60

    30

    11x

    9

    x

    60

    1616

    60

    x

    9 316

    3 8 32

    =

    1411

    5.52

    =

  • Examples

    To find the shorter leg from the longer leg:

    Find the value of x

    1.

    2.

    9

    x60

    10

    x

    30

    longer leg 3 longer leg3

    3 3 3

    =

    9x 3 3 3

    3= =

    10x 3

    3=