An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(1 ,4 )# to #(5 ,8 )# and the triangle's area is #27 #, 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 #(1  ,4   )# to #(5  ,8  )# and the triangle's area is #27  #, what are the possible coordinates of the triangle's third corner?

Wyatt Anderson · Accepted Answer

(9.75, -0.75) or (-3.75, 12.75)

First picture side A, common to two possible isosceles triangles, from (1,4) to (5,8). By Pythagoras we see that this side has length #4sqrt(2)#. Now draw the line perpendicular to side A which also bisects side A, up until it reaches where sides B and C of the isosceles triangle meet; this new line is the height of the triangle. Now you will realise that the area of the triangle, 27, is equal to #1/2bh = 1/2*4sqrt(2)*h#. Solving for #h#, we see that #h = (27sqrt(2))/4#. The final thing to note is that since this line is perpendicular to side A, which has a gradient of 1, this line has a gradient of -1. Hence, the horizontal and vertical components of the triangle connecting the midpoint of A and the other point of the isosceles triangle are equal in magnitude. Hence we can set up the equation #sqrt(2x^2)=(27sqrt(2))/4#, which when solved gives #x=+-27/4.# Now all that is left is to add this value of #x# to the midpoint #(3,6)# of side A, taking care with the signs, to obtain the possible coordinates of the final point. The point of the possible triangle to the bottom right of the midpoint is #(3+27/4,6-27/4) = (9.75, -0.75)# and the point to the top left of the midpoint is #(3-27/4, 6+27/4) = (-3.75, 12.75)#, both as required.

Maverick Smith · Answer

undefined To find the possible coordinates of the triangle's third corner, we can first calculate the length of side A using the distance formula. Then, we can use the formula for the area of a triangle to find the altitude corresponding to side A. Once we have the altitude, we can determine the direction in which it extends from the line containing side A. This will help us find the coordinates of the third corner.

Given the coordinates (1, 4) and (5, 8) for side A, the length of side A is:

$$ \sqrt{(5 - 1)^2 + (8 - 4)^2} = \sqrt{16 + 16} = \sqrt{32} $$

Since the triangle's area is given as 27, we can use the formula for the area of a triangle:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{altitude} $$

Solving for the altitude:

$$ 27 = \frac{1}{2} \times \sqrt{32} \times \text{altitude} $$

$$ \text{altitude} = \frac{27 \times 2}{\sqrt{32}} = \frac{54}{\sqrt{32}} $$

To find the direction of the altitude, we calculate the slope of side A:

$$ \text{Slope} = \frac{8 - 4}{5 - 1} = \frac{4}{4} = 1 $$

The negative reciprocal of the slope of side A gives the slope of the altitude:

$$ \text{Slope of altitude} = -\frac{1}{\text{Slope of side A}} = -\frac{1}{1} = -1 $$

Now, using the point-slope form of a line, we can find the equation of the line containing side A:

$$ y - 4 = 1(x - 1) $$

$$ y = x + 3 $$

Since the slope of the altitude is -1, the equation of the line perpendicular to side A passing through the midpoint of side A is:

$$ y - 6 = -1(x - 3) $$

$$ y = -x + 9 $$

Solving the system of equations formed by the line containing side A and the perpendicular line, we get the coordinates of the third corner:

$$ x + 3 = -x + 9 $$

$$ 2x = 6 $$

$$ x = 3 $$

Substituting $ x = 3 $ into either equation, we find:

$$ y = 3 + 3 = 6 $$

So, the possible coordinates of the triangle's third corner are (3, 6).