geometry chapter 7 · 30°−60°−90°theorem theorem 7-9: in a 30°−60°−90°right triangle,...
TRANSCRIPT
![Page 1: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/1.jpg)
Geometry Chapter 7
7-4: SPECIAL RIGHT TRIANGLES
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Warm-Up
Simplify the following.
1.) 10 × 30 2.) 45
5
3.) 88
84.) 3 × 27
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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
![Page 4: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/4.jpg)
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
![Page 5: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/5.jpg)
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
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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
![Page 7: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/7.jpg)
45° − 45° − 90° Examples
Find the value of x.
![Page 8: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/8.jpg)
45° − 45° − 90° Examples
Find the value of x.
Hypotenuse
Leg
45°
![Page 9: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/9.jpg)
45° − 45° − 90° Examples
Find the value of x.
Hypotenuse
LegSolution:
𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2
𝑥 = 12 × 2
𝒙 = 𝟏𝟐 𝟐
45°
![Page 10: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/10.jpg)
45° − 45° − 90° Examples
Find the value of x.
![Page 11: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/11.jpg)
45° − 45° − 90° Examples
Find the value of x.
Hypotenuse
Leg
Solution:
𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2
8 = x × 2
45°
![Page 12: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/12.jpg)
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°
![Page 13: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/13.jpg)
45° − 45° − 90° Examples
Find the values of x and y.
![Page 14: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/14.jpg)
45° − 45° − 90° Examples
Find the value of x and y.
Hypotenuse
Leg
45°
Leg
![Page 15: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/15.jpg)
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
![Page 16: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/16.jpg)
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
![Page 17: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/17.jpg)
45° − 45° − 90° Examples
Find the value of x.
𝟖
𝟖
𝒙
![Page 18: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/18.jpg)
45° − 45° − 90° Examples
Find the value of x.
𝟖
𝟖
𝒙
Hypotenuse
Leg
Leg
![Page 19: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/19.jpg)
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
![Page 20: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/20.jpg)
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
![Page 21: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/21.jpg)
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
![Page 22: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/22.jpg)
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
![Page 23: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/23.jpg)
30° − 60° − 90° Examples
Find the values of x and y.
![Page 24: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/24.jpg)
30° − 60° − 90° Examples
Find the values of x and y.
Shorter Leg
Hypotenuse
Longer Leg
![Page 25: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/25.jpg)
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
![Page 26: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/26.jpg)
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
![Page 27: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/27.jpg)
30° − 60° − 90° Examples
Find the values of x and y.
𝒙
𝒚
𝟐𝟎
𝟔𝟎°
![Page 28: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/28.jpg)
30° − 60° − 90° Examples
Find the values of x and y.
𝒙
𝒚
𝟐𝟎
𝟔𝟎°Shorter Leg
Hypotenuse
Longer Leg
![Page 29: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/29.jpg)
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
![Page 30: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/30.jpg)
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
![Page 31: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/31.jpg)
30° − 60° − 90° Examples
Find the values of x and y.
![Page 32: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/32.jpg)
30° − 60° − 90° Examples
Find the values of x and y.
Shorter Leg
Hypotenuse
Longer Leg
![Page 33: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/33.jpg)
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
![Page 34: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/34.jpg)
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
![Page 35: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/35.jpg)
30° − 60° − 90° Examples
Find the values of x and y.
𝟔𝟔
𝒙
𝟑𝟑
![Page 36: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/36.jpg)
30° − 60° − 90° Examples
Find the values of x and y.
𝟔𝟔
𝒙
𝟑𝟑Shorter Leg
Hypotenuse
Longer Leg
![Page 37: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/37.jpg)
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
![Page 38: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/38.jpg)
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
![Page 39: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/39.jpg)
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
![Page 40: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/40.jpg)
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
![Page 41: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/41.jpg)
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
![Page 42: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/42.jpg)
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
![Page 43: Geometry Chapter 7 · 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 3times](https://reader033.vdocuments.mx/reader033/viewer/2022042918/5f5ebfc11dcce660c0658835/html5/thumbnails/43.jpg)
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