Two corners of an isosceles triangle are at #(8 ,3 )# and #(5 ,4 )#. If the triangle's area is #15 #, what are the lengths of the triangle's sides?

Question

Two corners of an isosceles triangle are at #(8  ,3   )# and #(5   ,4   )#.  If the triangle's area is #15   #, what are the lengths of the triangle's sides?

Sadie Adams · Accepted Answer

#sqrt(10),5sqrt(3.7), 5sqrt(3.7)~=3.162,9.618,9.618#

The length of the given side is
#s=sqrt((5-8)^2+(4-3)^2)=sqrt(9+1)=sqrt(10)~=3.162#

From the formula of the triangle's area:
#S=(b*h)/2# => #15=(sqrt(10)*h)/2# => #h=30/sqrt(10)~=9.487#

Since the figure is an isosceles triangle we could have Case 1 , where the base is the singular side, ilustrated by Fig. (a) below

Or we could have Case 2 , where the base is one of the equal sides, ilustrated by Figs. (b) and (c) below

Sadie Adams · Answer

undefined To find the lengths of the sides of the isosceles triangle, we first need to find the coordinates of the third vertex. Then, we can calculate the lengths of the sides using the distance formula between the vertices.

Let's denote the coordinates of the third vertex as (x, y). Since the triangle is isosceles, the third vertex lies on the line of symmetry, which is the perpendicular bisector of the base.

We can find the equation of the perpendicular bisector using the midpoint formula and the negative reciprocal of the slope of the base. Then, we solve this equation simultaneously with the equation of the line passing through the given points to find the coordinates of the third vertex.

Once we have the coordinates of the third vertex, we can calculate the lengths of the sides using the distance formula.

After finding the lengths of the sides, we can verify if the triangle has an area of 15 square units using Heron's formula or other methods.