Geometry Chapter 7 · 30°−60°−90°Theorem Theorem 7-9: In a 30°−60°−90°right triangle,...
Transcript of Geometry Chapter 7 · 30°−60°−90°Theorem Theorem 7-9: In a 30°−60°−90°right triangle,...
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
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
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