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

Question

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

Mason Adams · Accepted Answer

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

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#.
The length of #b# is
#color(white)("XXX")abs(b)=sqrt((4-1)^2+(2-3)^2)=sqrt(10)~~3.162#

If #h# is the altitude of the triangle relative to base #b#
and given that the area is 2 (sq.units)
#color(white)("XXX")abs(h)=(2xx"Area")/abs(b)=4/sqrt(10)~~1.265#

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

Case A: #b# is not one of the equal sides of the isosceles triangle.

(using the previously determined values for #abs(h)# and #abs(b)#)

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

Case B: #b# is one of the equal sides of the isosceles triangle.

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

Since #abs(x+y)=abs(b)~~3.162#
and #abs(h)~~1.265#
(see prologue)

#color(white)("XXX")abs(y)~~sqrt(3.162^2-1.265^2)~~2.898#

#color(white)("XXX")abs(x)=abs(x+y)-abs(y)#
#color(white)("XXXX")=abs(b)-abs(y)#
#color(white)("XXXX") ~~3.162-2.898~~0.264#

and
#color(white)("XXX")abs(s)=sqrt(abs(h)^2+abs(x)^2)=sqrt(1.265^2+0.264^2)~~1.292#

Autumn Armstrong · Answer

undefined To find the lengths of the sides of the isosceles triangle:
1. Calculate the distance between the given points to determine the base of the triangle.
2. Use the formula for the area of a triangle ($ A = \frac{1}{2} \times \text{base} \times \text{height} $) to find the height of the triangle.
3. Once you have the base and the height, you can use the Pythagorean theorem to find the length of the other two sides since the triangle is isosceles and has two equal sides.