Answer:
Since MATH is a square, its diagonals MT and AH intersect at their midpoint, which is also the center of the square. Therefore, we can find the coordinates of the center by finding the intersection of the diagonal MT with its perpendicular bisector.
To do this, we first need to find the slope of the diagonal MT. We can rearrange the given equation into slope-intercept form, y = (-3/5)x + (12/5), where the slope is -3/5. Therefore, the slope of the perpendicular bisector of MT is the negative reciprocal of -3/5, which is 5/3.
Next, we can use the midpoint formula to find the coordinates of the center of the square. Since the diagonals of a square bisect each other, the midpoint of MT is also the midpoint of AH. Let (x,y) be the coordinates of the center. Then the midpoint formula gives:
(x+2)/2 = x/2 + 3/5(y-7/2)
(y+4)/2 = y/2 - 5/3(x-2)
Simplifying these equations and solving for x and y, we get:
x = 13/4
y = -1/4
Now that we know the coordinates of the center, we can use them to find the equation of the diagonal AH. Since A is (2,7), we know that AH is perpendicular to MT and passes through A. Therefore, its slope is the negative reciprocal of 5/3, which is -3/5. Using the point-slope form of the equation of a line, we get:
y - 7 = (-3/5)(x - 2)
y - 7 = (-3/5)x + 6/5
3x + 5y - 62 = 0
Therefore, the equation of the diagonal AH, in standard form, is 3x + 5y - 62 = 0.