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?

Question

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?

Aurora Ayala · Accepted Answer

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

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:

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#

Avery Allen · Answer

undefined To find the possible coordinates of the triangle's third corner, we first need to find the length of sides B and C, which are equal in length. Then, we can use the formula for the area of a triangle to find the height corresponding to side A. Once we have the height, we can find the possible coordinates of the third corner.

1. Calculate the length of side A:
   Use the distance formula to find the length of side A between the given points (2, 5) and (8, 7).

2. Use the formula for the area of a triangle:
   $ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $
   Plug in the length of side A and the given area to solve for the height.

3. Once you have the height, you can use it to find the possible coordinates of the third corner.
   - If side A is the base, the height is perpendicular to it and intersects the midpoint of side A.
   - Since the triangle is isosceles, the third corner lies on the perpendicular bisector of side A.

4. Calculate the equation of the perpendicular bisector of side A.
   - The midpoint of side A is the midpoint between the given points (2, 5) and (8, 7).
   - The slope of the perpendicular bisector is the negative reciprocal of the slope of side A.

5. Use the calculated equation of the perpendicular bisector to find the possible coordinates of the third corner by substituting different x-values and solving for y.

Repeat the process for both sides B and C to find all possible coordinates of the third corner.