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Geometry Chapter 7 7-4: SPECIAL RIGHT TRIANGLES
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Transcript of Geometry Chapter 7rgeometry16-17.weebly.com/uploads/8/6/1/2/86123628/...45°−45°−90°Triangles...

  • Geometry Chapter 7

    7-4: SPECIAL RIGHT TRIANGLES

  • Warm-Up

    Simplify the following.

    1.) 10 × 30 2.) 45

    5

    3.) 88

    84.) 3 × 27

  • Special Right Triangles

    Objective: Students will be able to use the relationships amongst the

    sides in special right triangles to find side lengths.

    Agenda

    45° − 45° − 90° Triangles

    30° − 60° − 90° Triangles

    Examples

  • 45° − 45° − 90° Triangles

    Definition

    A 45° − 45° − 90° Triangle is an isosceles right Triangle, with 45° as the measures of both the other two angles.

    45°

    45°

    Hypotenuse

    Leg

    Leg

  • 45° − 45° − 90° Triangles

    Definition

    A 45° − 45° − 90° Triangle is an isosceles right Triangle, with 45° as the measures of both the other two angles.

    Knowledge Connection

    Both Legs in this triangle are congruent.

    45°

    45°

    Hypotenuse

    Leg

    Leg

  • 45° − 45° − 90° Theorem

    Theorem 7.8: In a 45° − 45° − 90° right triangle, the hypotenuse is 2times as long as a leg.

    𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

    45°

    45°

    𝒄𝒂

    𝒃

    Hypotenuse

    Leg

    Leg

  • 45° − 45° − 90° Examples

    Find the value of x.

  • 45° − 45° − 90° Examples

    Find the value of x.

    Hypotenuse

    Leg

    45°

  • 45° − 45° − 90° Examples

    Find the value of x.

    Hypotenuse

    LegSolution:

    𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

    𝑥 = 12 × 2

    𝒙 = 𝟏𝟐 𝟐

    45°

  • 45° − 45° − 90° Examples

    Find the value of x.

  • 45° − 45° − 90° Examples

    Find the value of x.

    Hypotenuse

    Leg

    Solution:

    𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

    8 = x × 2

    45°

  • 45° − 45° − 90° Examples

    Find the value of x.

    Hypotenuse

    Leg

    Solution:

    𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

    8 = x × 2

    𝑥 =8

    2

    2=8 2

    2

    𝒙 = 𝟒 𝟐

    45°

  • 45° − 45° − 90° Examples

    Find the values of x and y.

  • 45° − 45° − 90° Examples

    Find the value of x and y.

    Hypotenuse

    Leg

    45°

    Leg

  • 45° − 45° − 90° Examples

    Find the value of x and y.

    Hypotenuse

    Leg

    For x:

    𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

    2 6 = x × 2

    𝑥 =2 6

    2

    𝒙 = 𝟐 𝟑

    45°

    Leg

  • 45° − 45° − 90° Examples

    Find the value of x and y.

    Hypotenuse

    Leg

    For x:

    𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

    2 6 = x × 2

    𝑥 =2 6

    2

    𝒙 = 𝟐 𝟑

    45°

    Leg

    For y:

    In a 45° − 45° − 90°triangle, the Legs have

    the same length.

    Therefore, 𝒚 = 𝟐 𝟑

  • 45° − 45° − 90° Examples

    Find the value of x.

    𝟖

    𝟖

    𝒙

  • 45° − 45° − 90° Examples

    Find the value of x.

    𝟖

    𝟖

    𝒙

    Hypotenuse

    Leg

    Leg

  • 45° − 45° − 90° Examples

    Find the value of x.

    For x:

    𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

    𝑥 = 8 × 2

    𝒙 = 𝟖 𝟐𝟖

    𝟖

    𝒙

    Hypotenuse

    Leg

    Leg

  • 30° − 60° − 90° Triangles

    Definition

    A 30° − 60° − 90° is a right triangle with 30°and 60° as its other angle measures.

    Shorter Leg

    Longer Leg

    Hypotenuse

  • 30° − 60° − 90° Triangles

    Definition

    A 30° − 60° − 90° is a right triangle with 30°and 60° as its other angle measures.

    Knowledge Connection

    The leg Opposite the 30° angle is called the Shorter Leg.

    The Leg Opposite the 60° angle is called the Longer Leg. Shorter Leg

    Longer Leg

    Hypotenuse

  • 30° − 60° − 90° Theorem

    Theorem 7-9: In a 30° − 60° − 90° right triangle, the hypotenuse is

    twice as long as the shorter leg, and the longer leg is 3 times as long as a shorter leg.

    𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

    𝐿. 𝐿. = 𝑆. 𝐿. × 3

    𝒄

    𝒂

    𝒃

    Shorter Leg

    Longer Leg

    Hypotenuse

  • 30° − 60° − 90° Examples

    Find the values of x and y.

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    Shorter Leg

    Hypotenuse

    Longer Leg

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    Shorter Leg

    Hypotenuse

    Longer Leg

    For x:𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

    𝑥 = 6 × 2

    𝒙 = 𝟏𝟐

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    Shorter Leg

    Hypotenuse

    Longer Leg

    For x:𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

    𝑥 = 6 × 2

    𝒙 = 𝟏𝟐

    For y:

    𝐿. 𝐿. = 𝑆. 𝐿. × 3

    𝑦 = 6 × 3

    𝒚 = 𝟔 𝟑

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    𝒙

    𝒚

    𝟐𝟎

    𝟔𝟎°

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    𝒙

    𝒚

    𝟐𝟎

    𝟔𝟎°Shorter Leg

    Hypotenuse

    Longer Leg

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    𝒙

    𝒚

    𝟐𝟎

    𝟔𝟎°Shorter Leg

    Hypotenuse

    Longer Leg For x:

    𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

    20 = 2x

    𝒙 = 𝟏𝟎

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    𝒙

    𝒚

    𝟐𝟎

    𝟔𝟎°Shorter Leg

    Hypotenuse

    Longer Leg For x:

    𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

    20 = 2x

    𝒙 = 𝟏𝟎

    For y:

    𝐿. 𝐿. = 𝑆. 𝐿. × 3

    𝑦 = 10 × 3

    𝒚 = 𝟏𝟎 𝟑

  • 30° − 60° − 90° Examples

    Find the values of x and y.

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    Shorter Leg

    Hypotenuse

    Longer Leg

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    Shorter Leg

    Hypotenuse

    Longer Leg

    For x:

    𝐿. 𝐿. = 𝑆. 𝐿. × 3

    8 = x 3

    𝑥 =8

    3

    𝑥 =8

    3

    3=𝟖 𝟑

    𝟑

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    Shorter Leg

    Hypotenuse

    Longer Leg

    For x:

    𝐿. 𝐿. = 𝑆. 𝐿. × 3

    8 = x 3

    𝑥 =8

    3

    𝑥 =8

    3∗

    3

    3=𝟖 𝟑

    𝟑

    For y:𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

    𝑦 = 𝑥 × 2

    𝑦 = 2 ×8 3

    3

    𝒚 =𝟏𝟔 𝟑

    𝟑

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    𝟔𝟔

    𝒙

    𝟑𝟑

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    𝟔𝟔

    𝒙

    𝟑𝟑Shorter Leg

    Hypotenuse

    Longer Leg

  • 30° − 60° − 90° Examples

    Find the values of x and y.

    𝟔𝟔

    𝒙

    𝟑𝟑

    For x:

    𝐿. 𝐿. = 𝑆. 𝐿. × 3

    x = 3 × 3

    𝒙 = 𝟑 𝟑

    Shorter Leg

    Hypotenuse

    Longer Leg

  • Final Practice: Both Triangles

    Find the values of the variables in the given diagram.

    For u:

    𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

    8 2 = u × 2

    𝑢 =8 2

    2

    𝒖 = 𝟖

    For v:

    In a 45° − 45° − 90°triangle, the Legs have

    the same length.

    Therefore, 𝐯 = 𝟖

  • Final Practice: Both Triangles

    Find the values of the variables in the given diagram.

    𝒏

    𝒎

    𝟏𝟎𝟒𝟓°

    For m:

    𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

    10 = m× 2

    𝑚 =10

    2

    𝒎 = 𝟓

    For n:

    In a 45° − 45° − 90°triangle, the Legs have

    the same length.

    Therefore, 𝐧 = 𝟓

  • Final Practice: Both Triangles

    Find the values of the variables in the given diagram.

    For a:

    𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

    𝑎 = 2 2 × 2

    𝑎 = 2(2)

    𝒂 = 𝟒

    For b:

    In a 45° − 45° − 90°triangle, the Legs have

    the same length.

    Therefore, 𝐛 = 𝟐 𝟐

  • Final Practice: Both Triangles

    Find the values of the variables in the given diagram.

    For u:

    𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

    u = 2 × 2

    𝒖 = 𝟒

    For v:

    𝐿. 𝐿. = 𝑆. 𝐿. × 3

    𝑦 = 2 × 3

    𝒚 = 𝟐 𝟑

  • Final Practice: Both Triangles

    Find the values of the variables in the given diagram.

    For y:

    𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

    8 5 = 2y

    𝒚 = 𝟒 𝟓

    For y:

    𝐿. 𝐿. = 𝑆. 𝐿. × 3

    𝑦 = 4 5 × 3

    𝒚 = 𝟒 𝟏𝟓

  • Final Practice: Both Triangles

    Find the values of the variables in the given diagram.

    For a:

    𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

    a = 11 × 2

    𝒂 = 𝟐𝟐

    For b:

    𝐿. 𝐿. = 𝑆. 𝐿. × 3

    11 3 = 𝑏 × 3

    𝑏 =11 3

    3

    𝒃 = 𝟏𝟏