Two corners of an isosceles triangle are at $(1, 3)$ $(1 ,3 )$ and $(9, 7)$ $(9 ,7 )$ . 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 $(1, 3)$ $(1 ,3 )$ and $(9, 7)$ $(9 ,7 )$ . If the triangle's area is $64$ $64$ , what are the lengths of the triangle's sides?

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Answer 1 · 2016-12-10T21:46:42

The sides of the triangle are $a = c = 15 and b = \sqrt{80}$ $a = c = 15 and b = sqrt(80)$

Explanation:

Let the length of side b equal the distance between the two given points:

$b = \sqrt{{(9 - 1)}^{2} + {(7 - 3)}^{2}}$ $b = sqrt((9 - 1)^2 + (7 - 3)^2)$

$b = \sqrt{{(8)}^{2} + {(4)}^{2}}$ $b = sqrt((8)^2 + (4)^2)$

$b = \sqrt{80}$ $b = sqrt(80)$

$A r e a = \frac{1}{2} b h$ $Area = 1/2bh$

$2 A r e a = b h$ $2Area = bh$

$h = \frac{2 A r e a}{b}$ $h = (2Area)/b$

$h = \frac{2 (64)}{\sqrt{80}}$ $h = (2(64))/sqrt(80)$

$h = \frac{128}{\sqrt{80}}$ $h = 128/sqrt(80)$

If side b is NOT one of the equal sides then the height is one of the legs of a right triangle and half of the length side b, $\frac{\sqrt{80}}{2}$ $sqrt(80)/2$ is the other leg. Therefore, we can use the Pythagorean Theorem to find the length of hypotenuse and this will be one of the equal sides:

$c = \sqrt{{(\frac{128}{\sqrt{80}})}^{2} + {(\frac{\sqrt{80}}{2})}^{2}}$ $c = sqrt((128/sqrt(80))^2 + (sqrt(80)/2)^2)$

$c \approx 15$ $c ~~ 15$

We need to find whether a triangle with sides, $a = c = 15 and b = \sqrt{80}$ $a = c = 15 and b = sqrt(80)$ has an area of 64.

I used a Heron's Formula Calculator and discovered that the area is 64.

The sides of the triangle are $a = c = 15 and b = \sqrt{80}$ $a = c = 15 and b = sqrt(80)$

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Two corners of an isosceles triangle are at $(1, 3)$ $(1 ,3 )$ and $(9, 7)$ $(9 ,7 )$ . If the triangle's area is $64$ $64$ , what are the lengths of the triangle's sides?

1 Answer

Explanation:

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Impact of this question

Science

Math

Humanities

... and beyond

Two corners of an isosceles triangle are at (1,3)(1 ,3 ) and (9,7)(9 ,7 ). If the triangle's area is 6464 , what are the lengths of the triangle's sides?

1 Answer

Explanation:

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Impact of this question

Two corners of an isosceles triangle are at $(1, 3)$ $(1 ,3 )$ and $(9, 7)$ $(9 ,7 )$ . If the triangle's area is $64$ $64$ , what are the lengths of the triangle's sides?