Two corners of an isosceles triangle are at #(8 ,5 )# and #(6 ,7 )#. 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  ,5   )# and #(6   ,7   )#.  If the triangle's area is #15   #, what are the lengths of the triangle's sides?

Mason Adams · Accepted Answer

Sides:#{2.8284, 10.7005,10.7005}#

Side #color(red)(a)# from #(8,5)# to #(6,7)#
has a length of
#color(red)(abs(a))=sqrt((8-6)^2+(5-7)^2)=2sqrt(2)~~2.8284#

Not that #color(red)(a)# can not be one of the equal length sides of the equilateral triangle since the maximum area such a triangle could have would be #(color(red)(2sqrt(2)))^2/2# which is less than #15#

Using #color(red)(a)# as the base and #color(blue)(h)# as the height relative to that base, we have
#color(white)("XXX")(color(red)(2sqrt(2))*color(blue)(h))/2 = color(brown)(15)#

#color(white)("XXX")rarr color(blue)(h) = 15/sqrt(2)#

Using the Pythagorean Theorem:
#color(white)("XXX")color(red)(b)=sqrt((15/sqrt(2))^2+((2sqrt(2))/2)^2) ~~10.70047#

and since the triangle is isosceles
#color(white)("XXX")c=b#

Aurora Ayala · Answer

undefined To find the lengths of the sides of the isosceles triangle, you can follow these steps:

1. Calculate the distance between the two given points to find the base length of the triangle.
2. Find the midpoint of the base to determine the height of the triangle.
3. Use the formula for the area of a triangle ($A = \frac{1}{2} \times \text{base} \times \text{height}$) to solve for the length of the sides.

Let's go through each step:

1. Distance between the two given points:
$$ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

2. Midpoint of the base:
$$ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) $$

3. Substitute the values into the formula for the area of the triangle:
$$ 15 = \frac{1}{2} \times \text{base} \times \text{height} $$

4. Solve for the base using the distance formula and the height using the midpoint formula.

5. Once you have the base and height, you can use the Pythagorean theorem to find the length of the two equal sides.

This process will give you the lengths of the sides of the isosceles triangle.