Two corners of an isosceles triangle are at $(3 ,9 )$ and $(2 ,7 )$ . If the triangle's area is $4$ , what are the lengths of the triangle's sides?

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Answer 1 · 2016-05-04T14:48:56

$color(brown)("As a simplified exact value:")$

$color(blue)(s=sqrt(549)/(2sqrt(17))=(3sqrt(1037))/34)$

$color(brown)("As an approximate decimal")$

$color(blue)(s~~2.831" to 3 decimal places")$

Tony B

Let the vertices be A,B and C
Let the corresponding sides be a, b, and c.

Let the width be w
Let the vertical height be h
Let the length of sides a and c be s

Given: Area = 4
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$color(blue)("Determine the value of w")$

Using Pythagoras $" "w=sqrt( (9-7)^2+(3-2)^2)$

$color(blue)(=> w= sqrt(16+1) = sqrt(17))$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$color(blue)("Determine the value of h")$

Given area $= 4 =1/2wh$

$color(blue)(h=8/w= 8/sqrt(17))$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Using Pythagoras

$s^2=(w/2)^2+h^2$

$s^2=(sqrt(17)/2)^2+(8/sqrt(17))^2$

$s=sqrt(17/4+64/17)$

$s=sqrt(545/68)$

$color(brown)("As a simplified exact value this:")$

$color(blue)(s=sqrt(549)/(2sqrt(17))=(3sqrt(1037))/34)$

$color(brown)("As an approximate decimal")$

$color(blue)(s~~2.831" to 3 decimal places")$

Two corners of an isosceles triangle are at (3 ,9 )(3 ,9 ) and (2 ,7 )(2 ,7 ). If the triangle's area is 4 4 , what are the lengths of the triangle's sides?