Question

if a rhombus has vertices at a(0,0), b(4,0), c(6,4), and d(2,4), which theorem proves that the diagonals are perpendicular?
a. pythagorean theorem
b. distance formula
c. midpoint formula
d. slope formula

what is the midpoint of the line segment connecting (1,2) and (5,10)?
a. (6,12)
b. (2,4)
c. (3,6)
d. (-3,-6)

what characteristic does a line have if its slope is zero?
a. the line is increasing
b. the line is horizontal
c. the line is decreasing
d. the line is vertical

Explanation:

Question 1:

To prove two lines (diagonals of a rhombus) are perpendicular, we use the slope - formula. If the product of the slopes of two lines is - 1, the lines are perpendicular. The Pythagorean theorem is for right - triangles, the distance formula is for finding the length between two points, and the mid - point formula is for finding the mid - point of a line segment.

Question 2:

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). For points \((1,2)\) and \((5,10)\), \(x_1=1,x_2 = 5,y_1=2,y_2 = 10\). Then \(\frac{1 + 5}{2}=\frac{6}{2}=3\) and \(\frac{2+10}{2}=\frac{12}{2}=6\). So the mid - point is \((3,6)\).

Question 3:

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). If \(m = 0\), then \(y_2-y_1=0\) (assuming \(x_2
eq x_1\)), which means the \(y\) - values of all points on the line are the same. A line with the same \(y\) - values for all \(x\) values is a horizontal line.

Answer:

1. d. Slope formula
2. c. \((3,6)\)
3. b. The line is horizontal