An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(2 ,5 )# to #(8 ,7 )# and the triangle's area is #12 #, what are the possible coordinates of the triangle's third corner?

1 Answer
Nov 17, 2016

The possible points are: #(3.8, 9.6)# and #(6.2, 2.4)#

Explanation:

The length of side "a" is:

#a = sqrt((8 -2)^2 + (7 - 5)^2) = sqrt(40) = 2sqrt(10)#

The area of a triangle is:

#"Area" = (1/2)"Base"xx"Height"#

Using side "a" as the base of the triangle and the given area, we can compute the height:

#12 = (1/2)2sqrt(10)xx"Height"#

#"Height" = 12/sqrt(10)#

Think of the height as the radius of a circle upon which the two possible points must lie:

#(x - h)^2 + (y - k)^2 = (12/sqrt(10))^2#

Because the triangle's other two sides are the same length, the center of this circle must be the midpoint between the points #(2, 5) and (8, 7):

#h = 2 + (8 - 2)/2 = 5 and k = 5 + (7 - 5)/2 = 6#

Substitute into the equation of the circle:

#(x - 5)^2 + (y - 6)^2 = (12/sqrt(10))^2" [1]"#

The two points must, also, lie on a line that is perpendicular to side "a"; the slope, m, of this line is:

#m = (2 - 8)/(7 - 5) = -6/2 = -3#

Using, this slope, the center point, and the point-slope form of the equation of a line, we write the equation upon which the two points must lie:

#y = -3(x - 5) + 6" [2]"#

Here is a graph of the equation [1], equation [2] and the two given points:

enter image source here

Substitute the right side of equation [2] into equation [1]:

#(x - 5)^2 + (-3(x - 5) + 6 - 6)^2 = (12/sqrt(10))^2#

#(x - 5)^2 + (-3(x - 5))^2 = (12/sqrt(10))^2#

#(x - 5)^2 + 9(x - 5)^2 = (12/sqrt(10))^2#

#(x - 5)^2 + 9(x - 5)^2 = 144/10#

#10(x - 5)^2 = 144/10#

#(x - 5)^2 = 144/100#

#x - 5 = +-12/10#

#x = 5 +-12/10#

#x = 3.8 and x = 6.2#

To obtain the corresponding y values, substitute these x values into equation [2]

#y = -3(3.8 - 5) + 6# and #y = -3(6.2 - 5) + 6#

#y = 9.6 and y = 2.4#