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# geometry

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Three equilateral triangles with side lengths 3,6 and 4, respectively, are joined as shown. Find the area of the shaded region.

Dec 21, 2019

#1
+109563
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Note that  in  both shaded triangles.....the  angles  that  are touching the base of the  figure  must   = 60°

So  the shaded area ( a triangle)  on the left  =  (1/2) (6)(3)sin(60°)

And   the shaded area on the right (a riangle)  = (1/2) * (6) (4) sin (60°)

So....the total area of the two shaded regions  =

(1/2)(6)sin(60°)  [ 3 + 4 ]  =

3(√3/2)  [ 7]  =

21√3

_____     units^2    ≈   18.19 units^2

2

Dec 21, 2019
edited by CPhill  Dec 21, 2019
edited by CPhill  Dec 21, 2019
#2
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I'm stuck trying to understand this.

1.  How do you tell that the shaded triangles are right triangles?

2.  It looks like the 6 that you're multiplying by is the hypotenuse of those shaded triangles instead of the height.

I'm sorry for being so dense.

.

Guest Dec 21, 2019
#4
+109563
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Sorry...I messed up......see  my edited answer above.....

CPhill  Dec 21, 2019
edited by CPhill  Dec 21, 2019
#3
+531
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The area of the triangle on the left is 7.794 units2, and the area of the triangle on the right is  10.394 units2.

Therefore, the total shaded area is:  18.188 units2

Dec 21, 2019