A triangle has sides A, B, and C. The angle between sides A and B is $(pi)/2$ and the angle between sides B and C is $pi/12$ . If side B has a length of 24, what is the area of the triangle?

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A triangle has sides A, B, and C. The angle between sides A and B is $(pi)/2$ and the angle between sides B and C is $pi/12$ . If side B has a length of 24, what is the area of the triangle?

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Answer 1 · 2017-05-22T02:25:29

$A=77.172$ $units^2$

Explanation:

$Acolor(white)(00)color(black)(pi/12)color(white)(0000000)acolor(white)(00)color(white)(0)$

$Bcolor(white)(00)color(white)(pi/1 3)color(white)(0000000)bcolor(white)(00)color(black)(24)$

$Ccolor(white)(00)color(black)(pi/2)color(white)(0000000)c color(white)(00)color(white)(0)$

Let's find the remaining angle, $B$ :

$pi-pi/12-pi/2$ leaves us with $(5pi)/12$

$Acolor(white)(00)color(black)(pi/12)color(white)(00000000)acolor(white)(00)color(white)(0)$

$Bcolor(white)(00)color(black)((5pi)/12)color(white)(.)color(white)(0000000)bcolor(white)(00)color(black)(24)$

$Ccolor(white)(00)color(black)(pi/2)color(white)(000000000)c color(white)(00)color(white)(0)$

Now we could use law of sines, but I noticed something... $pi/2$ is a right angle! Here's what our triangle looks like:

$color(white)(a)color(white)(- - - - - - - - - -)color(black)(/)color(black)(|)$
$color(white)(a)color(white)(- - - - - - - - -)color(black)(/)color(white)(-)color(black)(|)$
$color(white)(a)color(white)(- - - - - - - -)color(black)(/)color(white)(- - .)color(black)(|)$
$color(white)(a)color(white)(- - - - - - -)color(black)(/)color(white)(- - -0)color(black)(|)$
$color(white)(- - - -)color(black)(c)color(white)(00000)color(black)(/)color(white)(- - - -)color(white)(/)color(black)(|)color(black)(a)$
$color(white)(a)color(white)(- - - - -)color(black)(/)color(white)(- - - - -0)color(black)(|)$
$color(white)(a)color(white)(- - - -)color(black)(/)color(white)(- - - - - -0)color(black)(|)$
$color(white)(a)color(white)(- - -)color(black)(/)color(white)(- - - - - - -0)color(black)(|)$
$color(white)(a)color(white)(- -)color(black)(/)color(white)(- - - - - - - -0)color(black)(|)$
$color(white)(-)color(black)(/)color(black)()color(black)(A)color(black)(...........................................)$
$color(white)(0000000000000000) b$

What we need to find is the height, or $color(red)(a)$ and the base, or $color(red)(b)$

$Acolor(white)(00)color(black)(pi/12)color(white)(00000000)acolor(white)(00)color(white)(0)$

$Bcolor(white)(00)color(black)((5pi)/12)color(white)(.)color(white)(0000000)bcolor(white)(00)color(red)(24)$

$Ccolor(white)(00)color(black)(pi/2)color(white)(000000000)c color(white)(00)color(white)(0)$

To find length $color(red)(a)$ , we should use $tan$ :

$tan(A)=a/b$

$tan(pi/12)=a/24$

$tan(pi/12) xx 24 = a$

$a ~~ 6.431$

Now we can find the area:

$A=(a xx b)/2$

$A=(6.431 xx 24)/2$

$A=77.172$ $units^2$

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A triangle has sides A, B, and C. The angle between sides A and B is $(pi)/2$ and the angle between sides B and C is $pi/12$ . If side B has a length of 24, what is the area of the triangle?

1 Answer

Explanation:

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A triangle has sides A, B, and C. The angle between sides A and B is (pi)/2(pi)/2 and the angle between sides B and C is pi/12pi/12. If side B has a length of 24, what is the area of the triangle?

1 Answer

Explanation:

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A triangle has sides A, B, and C. The angle between sides A and B is $(pi)/2$ and the angle between sides B and C is $pi/12$ . If side B has a length of 24, what is the area of the triangle?