Two corners of an isosceles triangle are at $(8, 3)$ $(8 ,3 )$ and $(5, 4)$ $(5 ,4 )$ . If the triangle's area is $15$ $15$ , what are the lengths of the triangle's sides?

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Two corners of an isosceles triangle are at $(8, 3)$ $(8 ,3 )$ and $(5, 4)$ $(5 ,4 )$ . If the triangle's area is $15$ $15$ , what are the lengths of the triangle's sides?

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Answer 1 · 2016-02-09T20:35:10

$\sqrt{10}, 5 \sqrt{3.7}, 5 \sqrt{3.7} ≅ 3.162, 9.618, 9.618$ $sqrt(10),5sqrt(3.7), 5sqrt(3.7)~=3.162,9.618,9.618$

Explanation:

The length of the given side is
$s = \sqrt{{(5 - 8)}^{2} + {(4 - 3)}^{2}} = \sqrt{9 + 1} = \sqrt{10} ≅ 3.162$ $s=sqrt((5-8)^2+(4-3)^2)=sqrt(9+1)=sqrt(10)~=3.162$

From the formula of the triangle's area:
$S = \frac{b \cdot h}{2}$ $S=(b*h)/2$ => $15 = \frac{\sqrt{10} \cdot h}{2}$ $15=(sqrt(10)*h)/2$ => $h = \frac{30}{\sqrt{10}} ≅ 9.487$ $h=30/sqrt(10)~=9.487$

Since the figure is an isosceles triangle we could have Case 1 , where the base is the singular side, ilustrated by Fig. (a) below

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Or we could have Case 2 , where the base is one of the equal sides, ilustrated by Figs. (b) and (c) below

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For this problem Case 1 always applies, because:

$tan (\frac{α}{2}) = \frac{\frac{a}{2}}{h}$ $tan(alpha/2)=(a/2)/h$ => $h = (\frac{1}{2}) \frac{a}{tan (\frac{α}{2})}$ $h=(1/2)a/tan(alpha/2)$

But there's a condition so that Case 2 applies:

$sin (β) = \frac{h}{b}$ $sin(beta)=h/b$ => $h = b sin β$ $h=bsin beta$
Or $h = b sin γ$ $h=bsin gamma$
Since the highest value of $sin β$ $sin beta$ or $sin γ$ $sin gamma$ is $1$ $1$ , the highest value of $h$ $h$ , in Case 2, must be $b$ $b$ .

In the present problem h is longer than the side to which it is perpendicular, so for this problem only the Case 1 applies.

Solution considering Case 1 (Fig. (a))

$b^{2} = h^{2} + {(\frac{a}{2})}^{2}$ $b^2=h^2+(a/2)^2$
$b^{2} = {(\frac{30}{\sqrt{10}})}^{2} + {(\frac{\sqrt{10}}{2})}^{2}$ $b^2=(30/sqrt(10))^2+(sqrt(10)/2)^2$
$b^{2} = \frac{900}{10} + \frac{10}{4} = \frac{900 + 25}{10} = \frac{925}{10}$ $b^2=900/10+10/4=(900+25)/10=925/10$ => $b = \sqrt{92.5} = 5 \sqrt{3.7} ≅ 9.618$ $b=sqrt(92.5)=5sqrt(3.7)~=9.618$

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Two corners of an isosceles triangle are at $(8, 3)$ $(8 ,3 )$ and $(5, 4)$ $(5 ,4 )$ . If the triangle's area is $15$ $15$ , what are the lengths of the triangle's sides?

1 Answer

Explanation:

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Science

Math

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Two corners of an isosceles triangle are at (8,3)(8 ,3 ) and (5,4)(5 ,4 ). If the triangle's area is 1515 , what are the lengths of the triangle's sides?

1 Answer

Explanation:

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Two corners of an isosceles triangle are at $(8, 3)$ $(8 ,3 )$ and $(5, 4)$ $(5 ,4 )$ . If the triangle's area is $15$ $15$ , what are the lengths of the triangle's sides?