The coordinates of three corners of a square are (1, 7), (5, 3), and (9, 7). What are the coordinates of the fourth corner of the square? (1 point) Responses (5, 7) (5, 7) (5, 11) (5, 11) (9, 1) (9, 1) (13, 3)
The coordinates of the fourth corner of the square can be determined by finding the midpoint of the line connecting two of the given corners and then finding the reflection of this midpoint across the line connecting the remaining two given corners.
The midpoint of the line connecting (1, 7) and (5, 3) is:
Midpoint = ((1+5)/2, (7+3)/2) = (3, 5)
The line connecting (3, 5) and (9, 7) has a slope of (7-5)/(9-3) = 2/6 = 1/3.
The equation of this line can be found using the point-slope form:
y - 5 = (1/3)(x - 3)
Simplifying, we get:
3y - 15 = x - 3
x - 3y = -12
Now we want to find the reflection of (3, 5) across this line. The distance from (3, 5) to the line is given by the equation:
Distance = |x - 3y - (-12)| / sqrt(1^2 + (-3)^2) = |x - 3y + 12| / sqrt(10)
This distance must be the same as the distance between any of the given corners and the fourth corner. The distance between (1, 7) and the fourth corner is sqrt((x - 1)^2 + (y - 7)^2).
Using the distance formula, the equation becomes:
sqrt((x - 1)^2 + (y - 7)^2) = |x - 3y + 12| / sqrt(10)
Squaring both sides and simplifying, we get:
(x - 1)^2 + (y - 7)^2 = (x - 3y + 12)^2 / 10
Expanding and simplifying further, we get:
x^2 - 2x + 1 + y^2 - 14y + 49 = (x^2 - 6xy + 9y^2 + 24x - 72y + 144) / 10
Multiplying through by 10 to eliminate the fraction, we get:
10x^2 - 20x + 10 + 10y^2 - 140y + 490 = x^2 - 6xy + 9y^2 + 24x - 72y + 144
Simplifying and collecting like terms, we get:
9x^2 + 6xy - y^2 - 4x + 8y - 46 = 0
Thus, the equation of the line of reflection is:
9x^2 + 6xy - y^2 - 4x + 8y - 46 = 0
Now we can find the reflection point by solving the system of equations formed by the line of reflection and the equation of the original line connecting (3, 5) and (9, 7).
Solving these equations, we find that the coordinates of the fourth corner of the square are:
(5, 11)
Therefore, the correct answer is: (5, 11)
The idea is good, but since one diagonal is a horizontal segment from (1,7) to (9,7), it is clear that the midpoint is at (5,7). Thus, the reflection of (5,3) across that line is (5,11).
All those calculations for the general case did not need to be done for this problem.