An isosceles triangle has sides #a, b# and #c# with sides #b# and #c# being equal in length. If side #a# goes from #(2 ,9 )# to #(8 ,5 )# and the triangle's area is #64 #, 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  ,9  )# to #(8  ,5 )# and the triangle's area is #64  #, what are the possible coordinates of the triangle's third corner?

Sadie Allen · Accepted Answer

#(-63/13, -101/13) or (193/13, 283/13)#

Side #a# is #(2,9)# to #(8,5)#

Let's do this directly. We have

#a^2 =(8-2)^2+(5-9)^2=36+16=52#

Area #S = 1/2 a h # or

#h = {2S}/a = {2(64)}/\sqrt{52} = 128/sqrt{52}#

The other vertex is along the perpendicular bisector of side #a#. The direction vector for side as is #(8-2,5-9)=(6,-4)# so the perpendicular through the midpoint is

#(x,y) = ( {2+8}/2, {9+5}/2) + t(4,6) = (5,7)+t(4,6)#

We need #|t(4,6)|=h#

#t = pm h/|(4.6)| = pm (128/sqrt{52})/sqrt{52} = pm 128/52 = pm 32/13#

Two choices for the third vertex:

#(x,y) = (5,7) pm (32/13) (4,6) = (-63/13, -101/13) or (193/13, 283/13)#

Let's check one:

# b^2 = (-63/13 - 2)^2 + (-101/13 - 9)^2 = 4265/13 #

# c^2 = (-63/13 - 8)^2 + (-101/13 - 5)^2 = 4265/13 quad sqrt#

Area #S#

#16S^2 = 4(52)(4265/13) - (52)^2 #

#S = 64 quad sqrt#

James Atkins · Answer

undefined To find the possible coordinates of the triangle's third corner, we first need to determine the lengths of sides $b$ and $c$, which are equal, using the given coordinates. Then we can use the formula for the area of a triangle to solve for the possible coordinates of the third corner.

1. Calculate the length of side $a$ using the distance formula:
$$ a = \sqrt{(8 - 2)^2 + (5 - 9)^2} $$

2. Since the triangle is isosceles, sides $b$ and $c$ have the same length as side $a$.

3. Given the area of the triangle ($64$), we can use the formula for the area of a triangle:
$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$
$$ 64 = \frac{1}{2} \times a \times h $$
$$ h = \frac{2 \times 64}{a} $$

4. Now, we can use the height and the midpoint of side $a$ to find the coordinates of the third corner.

5. Calculate the midpoint of side $a$:
$$ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

6. Find the equation of the line perpendicular to side $a$ passing through its midpoint:
$$ \text{Slope of } a = m_a = \frac{y_2 - y_1}{x_2 - x_1} $$
$$ \text{Slope of perpendicular line} = -\frac{1}{m_a} $$

7. Use the slope of the perpendicular line and the midpoint to find the equation of the line:
$$ y - y_{\text{midpoint}} = m_{\text{perpendicular}}(x - x_{\text{midpoint}}) $$

8. Substitute the calculated midpoint coordinates and slope into the equation and solve for $y$ to find the possible coordinates of the third corner.

9. Repeat the process for the negative slope of the perpendicular line to find the other possible coordinates.

By following these steps, you can determine the possible coordinates of the triangle's third corner.