Two corners of an isosceles triangle are at $(4, 2)$ $(4 ,2 )$ and $(1, 3)$ $(1 ,3 )$ . If the triangle's area is $2$ $2$ , what are the lengths of the triangle's sides?

Question

Two corners of an isosceles triangle are at $(4, 2)$ $(4 ,2 )$ and $(1, 3)$ $(1 ,3 )$ . If the triangle's area is $2$ $2$ , what are the lengths of the triangle's sides?

1 Answer

Impact of this question

1360 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-12T17:29:54

Sides:
$XXX {3.162, 2.025, 2.025}$ $color(white)("XXX"){3.162, 2.025, 2.025}$
or
$XXX {3.162, 3.162, 1.292}$ $color(white)("XXX"){3.162,3.162,1.292}$

Explanation:

There are two cases that need to be considered (see below).

For both cases I will refer to the line segment between the given point coordinates as $b$ $b$ .
The length of $b$ $b$ is
$XXX | b | = \sqrt{{(4 - 1)}^{2} + {(2 - 3)}^{2}} = \sqrt{10} \approx 3.162$ $color(white)("XXX")abs(b)=sqrt((4-1)^2+(2-3)^2)=sqrt(10)~~3.162$

If $h$ $h$ is the altitude of the triangle relative to base $b$ $b$
and given that the area is 2 (sq.units)
$XXX | h | = \frac{2 \times Area}{| b |} = \frac{4}{\sqrt{10}} \approx 1.265$ $color(white)("XXX")abs(h)=(2xx"Area")/abs(b)=4/sqrt(10)~~1.265$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Case A: $b$ $b$ is not one of the equal sides of the isosceles triangle.
enter image source here
Notice that the altitude $h$ $h$ divides the triangle into two right triangles.
If the equal sides of the triangle are denoted as $s$ $s$
then
$XXX | s | = \sqrt{{| h |}^{2} + {(\frac{| b |}{2})}^{2} \approx 2.025}$ $color(white)("XXX")abs(s)=sqrt(abs(h)^2+(abs(b)/2)^2~~2.025$
(using the previously determined values for $| h |$ $abs(h)$ and $| b |$ $abs(b)$ )

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Case B: $b$ $b$ is one of the equal sides of the isosceles triangle.
enter image source here

Note that the altitude, $h$ $h$ , divides $b$ $b$ into two sub-line segments which I have labelled $x$ $x$ and $y$ $y$ (see diagram above).

Since $| x + y | = | b | \approx 3.162$ $abs(x+y)=abs(b)~~3.162$
and $| h | \approx 1.265$ $abs(h)~~1.265$
(see prologue)

$XXX | y | \approx \sqrt{{3.162}^{2} - {1.265}^{2}} \approx 2.898$ $color(white)("XXX")abs(y)~~sqrt(3.162^2-1.265^2)~~2.898$

$XXX | x | = | x + y | - | y |$ $color(white)("XXX")abs(x)=abs(x+y)-abs(y)$
$XXXX = | b | - | y |$ $color(white)("XXXX")=abs(b)-abs(y)$
$XXXX \approx 3.162 - 2.898 \approx 0.264$ $color(white)("XXXX") ~~3.162-2.898~~0.264$

and
$XXX | s | = \sqrt{{| h |}^{2} + {| x |}^{2}} = \sqrt{{1.265}^{2} + {0.264}^{2}} \approx 1.292$ $color(white)("XXX")abs(s)=sqrt(abs(h)^2+abs(x)^2)=sqrt(1.265^2+0.264^2)~~1.292$

Science

Math

Humanities

... and beyond

Two corners of an isosceles triangle are at $(4, 2)$ $(4 ,2 )$ and $(1, 3)$ $(1 ,3 )$ . If the triangle's area is $2$ $2$ , what are the lengths of the triangle's sides?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

Two corners of an isosceles triangle are at (4,2)(4 ,2 ) and (1,3)(1 ,3 ). If the triangle's area is 22 , what are the lengths of the triangle's sides?

1 Answer

Explanation:

Related questions

Impact of this question

Two corners of an isosceles triangle are at $(4, 2)$ $(4 ,2 )$ and $(1, 3)$ $(1 ,3 )$ . If the triangle's area is $2$ $2$ , what are the lengths of the triangle's sides?