Two corners of an isosceles triangle are at $(9, 2)$ $(9 ,2 )$ and $(4, 7)$ $(4 ,7 )$ . If the triangle's area is $64$ $64$ , what are the lengths of the triangle's sides?

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Two corners of an isosceles triangle are at $(9, 2)$ $(9 ,2 )$ and $(4, 7)$ $(4 ,7 )$ . If the triangle's area is $64$ $64$ , what are the lengths of the triangle's sides?

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Answer 1 · 2016-11-02T22:04:26

Solution. $\frac{\sqrt[2]{34018}}{10} \approx 18.44$ $root2{34018}/10~~18.44$

Explanation:

Let's take the points $A (9; 2)$ $A(9;2)$ and $B (4; 7)$ $B(4;7)$ as the base vertices.
$A B = \sqrt[2]{{(9 - 4)}^{2} + {(2 - 7)}^{2}} = 5 \sqrt[2]{2}$ $AB=root2{(9-4)^2+(2-7)^2}=5root2{2}$ , the height $h$ $h$ can be taken out from formula of the area $5 \sqrt[2]{2} \cdot \frac{h}{2} = 64$ $5root2{2}*h/2=64$ . In such a way $h = 64 \cdot \frac{\sqrt[2]{2}}{5}$ $h=64*root2{2}/5$ .

The third vertex $C$ $C$ must be on the axis of $A B$ $AB$ that is the line perpendicular to $A B$ $AB$ passing through its medium point $M (\frac{13}{2}; \frac{9}{2})$ $M(13/2;9/2)$ .
This line is $y = x - 2$ $y=x-2$ and $C (x; x - 2)$ $C(x;x-2)$ .

$C M^{2} = {(x - \frac{13}{2})}^{2} + {(x - 2 - \frac{9}{2})}^{2} = h^{2} = 2^{12} \cdot \frac{2}{5^{2}}$ $CM^2=(x-13/2)^2+(x-2-9/2)^2=h^2=2^12*2/5^2$ .
It gets $x^{2} - 13 x + \frac{169}{4} - \frac{2^{12}}{25} = 0$ $x^2-13x+169/4-2^12/25=0$ that solved yelds to values possible for the third vertex, $C = (\frac{193}{10}, \frac{173}{10})$ $C=(193/10,173/10)$ or $C = (- \frac{63}{10}, - \frac{83}{10})$ $C=(-63/10,-83/10)$ .

The length of the equal sides is $A C = \sqrt[2]{{(9 - \frac{193}{10})}^{2} + {(2 - \frac{173}{10})}^{2}} = \sqrt[2]{{(\frac{103}{10})}^{2} + {(- \frac{153}{10})}^{2}} = \frac{\sqrt[2]{34018}}{10} \approx 18.44$ $AC=root2{(9-193/10)^2+(2-173/10)^2}=root2{(103/10)^2+(-153/10)^2}=root2{34018}/10~~18.44$

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Two corners of an isosceles triangle are at $(9, 2)$ $(9 ,2 )$ and $(4, 7)$ $(4 ,7 )$ . If the triangle's area is $64$ $64$ , what are the lengths of the triangle's sides?

1 Answer

Explanation:

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Science

Math

Humanities

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Two corners of an isosceles triangle are at (9,2)(9 ,2 ) and (4,7)(4 ,7 ). If the triangle's area is 6464 , what are the lengths of the triangle's sides?

1 Answer

Explanation:

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Two corners of an isosceles triangle are at $(9, 2)$ $(9 ,2 )$ and $(4, 7)$ $(4 ,7 )$ . If the triangle's area is $64$ $64$ , what are the lengths of the triangle's sides?