Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{2}$ $( pi ) / 2$ . If one side of the triangle has a length of $1$ $1$ , what is the longest possible perimeter of the triangle?

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Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{2}$ $( pi ) / 2$ . If one side of the triangle has a length of $1$ $1$ , what is the longest possible perimeter of the triangle?

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Answer 1 · 2016-06-30T18:42:34

$Perimeter \approx 6.03 to 2 decimal places$ $"Perimeter "~~6.03" to 2 decimal places"$

Explanation:

Method: assign the length of 1 to the shortest side. Consequently we need to identify the shortest side.

Tony B

Extend CA to point P

Let $∠ A C B = \frac{π}{2} \to 90^{0}$ $/_ACB=pi/2 ->90^0$ Thus triangle ABC is a right triangle.

That being so then $∠ C A B + ∠ A B C = \frac{π}{2} thus ∠ C A B < \frac{π}{2} and ∠ A B C < \frac{π}{2}$ $/_CAB+/_ABC = pi/2" thus " /_CAB < pi/2" and "/_ABC < pi/2$

Consequently the other given angle of magnitude $\frac{5}{8} π$ $5/8 pi$ has to an external angle

Let $∠ B A P = \frac{5}{8} π \Rightarrow ∠ C A B = \frac{3}{8} π$ $/_BAP=5/8 pi=>/_CAB=3/8 pi$

As $∠ C A B > ∠ A B C$ $/_CAB > /_ABC$ then AC < CB
Also as AC < AB and BC < AC, $AC is the shortest length$ $color(blue)("AC is the shortest length")$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given that AC = 1

Thus for $∠ C A B$ $/_CAB$

$A B cos (\frac{3}{8} π) = 1$ $ABcos(3/8 pi)=1$

$A B = \frac{1}{cos (\frac{3}{8} π)} \approx 2.6131 to 4 decimal places$ $color(blue)(AB = 1/cos(3/8 pi)~~2.6131" to 4 decimal places")$
'................................................................................

$tan (\frac{3}{8} π) = \frac{B C}{A C} = \frac{B C}{1} = B C \approx 2.4142 to 4 decimal places$ $color(blue)(tan(3/8 pi)=(BC)/(AC) = (BC)/1=BC~~2.4142" to 4 decimal places")$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Perimeter = $1 + \frac{1}{cos (\frac{3}{8} π)} + tan (\frac{3}{8} π)$ $1+ 1/cos(3/8 pi)+tan(3/8 pi)$

$\approx 6.0273 to 4 decimal places$ $~~6.0273" to 4 decimal places"$

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Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{2}$ $( pi ) / 2$ . If one side of the triangle has a length of $1$ $1$ , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Impact of this question

Science

Math

Humanities

... and beyond

Two corners of a triangle have angles of 5π8 (5 pi )/ 8 and π2 ( pi ) / 2 . If one side of the triangle has a length of 1 1 , what is the longest possible perimeter of the triangle?

1 Answer

Explanation:

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Impact of this question

Two corners of a triangle have angles of $\frac{5 π}{8}$ $(5 pi )/ 8$ and $\frac{π}{2}$ $( pi ) / 2$ . If one side of the triangle has a length of $1$ $1$ , what is the longest possible perimeter of the triangle?