An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(7 ,5 )# to #(8 ,1 )# and the triangle's area is #15 #, 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 #(7  ,5  )# to #(8  ,1 )# and the triangle's area is #15  #, what are the possible coordinates of the triangle's third corner?

Autumn Armstrong · Accepted Answer

#(15/34, 42/34)# and #(495/34, 162/34)#

We can find the length of 'a' by finding the distance between the two points:

#a = sqrt((7 - 8)² + (5 - 1)²)#

#a = sqrt((-1)² + (4)²)#

#a = sqrt(1 + 16)#

#a = sqrt(17)#

Let side 'a' be the base of the triangle.

Using the area, we can compute the height:

#A = (1/2)bh = (1/2)ah#

#15 = (1/2)(sqrt17)h#

#h = 30sqrt17/17#

The height must lie on the line that is the perpendicular bisector of side 'a'. Let's find the equation of that line:

Side 'a' goes from left to right 1 unit and down 4 units (For later use, remember this is slope, -4), therefore, the midpoint goes from left to right #1/2# unit and down 2 units.

The midpoint is #(15/2, 3)#

A perpendicular line will have a slope that is the negative reciprocal of -4:

#-1/-4 = 1/4#

Using the point-slope form of the equation of a line, #y-y_1 = m(x - x_1)#, we write an equation of a line upon which both possible vertices must lie:

#y - 3 = 1/4(x - 15/2)#

#y = 1/4x + 9/8#

Using the equation for a circle we write an equation where the radius is #r = h = 30sqrt17/17# and the center is the midpoint #(15/2, 3)#:

#(30sqrt17/17)² = (x - 15/2)² + (y - 3)²#

#900/17 = x² - 15x + 225/4 + y² -6y + 9#

Substitute #1/4x + 9/8# for y:

#900/17 = x² - 15x + 225/4 + (1/4x + 9/8)² -6(1/4x + 9/8) + 9#

I used Wolframalpha to solve this:

#x = 15/34# and #x = 495/34#

The corresponding y coordinates are:

#y = 42/34# and #y = 162/34#

Aurora Ayala · Answer

undefined To find the possible coordinates of the triangle's third corner, we need to determine the length of sides $ B $ and $ C $ of the isosceles triangle and then find the perpendicular bisector of side $ A $. The third corner can lie on this perpendicular bisector.

1. Calculate the length of side $ A $ using the distance formula.
2. Since sides $ B $ and $ C $ are equal in length, the midpoint of side $ A $ is equidistant from both endpoints of side $ A $. This midpoint is a point on the perpendicular bisector of side $ A $.
3. Determine the equation of the line passing through the midpoint of side $ A $ and perpendicular to side $ A $. This line represents the perpendicular bisector of side $ A $.
4. Find the intersection of the perpendicular bisector with the line passing through the given endpoints of side $ A $. These points represent the possible coordinates of the triangle's third corner.

By considering the intersection of the perpendicular bisector with the line passing through the given endpoints of side $ A $, you'll find the potential coordinates of the triangle's third corner.