The referenced figure is below. 

The two non-horizontal line segments are perpendicular, as is proved as follows:

The slope of the line that connects \(\displaystyle (12, 0 )\) and \(\displaystyle (0,12)\) can be found using the slope formula, setting \(\displaystyle x_{1} = 12 , y_{2} = x_{2} = 0, y_{2} = 12\):

\(\displaystyle m_{1} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}= \frac{12-0}{0-12} = \frac{12}{-12} = -1\)

The slope of the line that connects \(\displaystyle (12, 0 )\) and \(\displaystyle (0,-12)\) can be found similarly, setting \(\displaystyle x_{1} = 12 , y_{2} = x_{2} = 0, y_{2} = -12\):

\(\displaystyle m_{2} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}= \frac{-12-0}{0-12} = \frac{-12}{-12} = 1\)

The product of their slopes is \(\displaystyle m_{1} m_{2} = -1 \cdot 1 = -1\), which indicates perpendicularity between the sides.

This makes the triangle right, and the side with endpoints \(\displaystyle (12, 0 )\) and \(\displaystyle (0,-12)\) the hypotenuse. The center of the circle that circumscribes a right triangle is the midpoint of its hypotenuse, which is easily be seen to be the origin, \(\displaystyle (0,0)\).

The referenced figure is below. 

The triangle formed is a right triangle whose hypotenuse is the segment with the endpoints \(\displaystyle (0, 8)\) and \(\displaystyle (12,0)\). The center of the circle that circumscribes a right triangle is the midpoint of its hypotenuse, so the midpoint formula 

\(\displaystyle (x_{M}, y_{M}) = \left ( \frac{x_{1}+ x_{2} }{2},\frac{y_{1}+ y_{2} }{2} \right )\)

can be applied, setting \(\displaystyle x_{1} = 12, y_{1} = x_{2} = 0, y_{2} = 8\):

\(\displaystyle x_{M}= \frac{x_{1}+ x_{2} }{2}= \frac{12+0}{2}= \frac{12}{2} = 6\)

\(\displaystyle y_{M}= \frac{y_{1}+ y_{2} }{2}= \frac{0+8}{2}= \frac{8}{2} = 4\)

The midpoint of the hypotenuse, and, consequently, the center of the circumscribed circle, is the point with coordinates \(\displaystyle (6, 4)\).

Another way of viewing this problem is to note that the three given vertices form a triangle \(\displaystyle \bigtriangleup XYZ\) whose sides' perpendicular bisectors intersect at the points \(\displaystyle A\), \(\displaystyle B\), and \(\displaystyle C\). However, the three perpendicular bisectors of the sides of any triangle always intersect at a common point. The correct response is that \(\displaystyle A\), \(\displaystyle B\), and \(\displaystyle C\) are the same point.

The length of the horizontal leg of the triangle is the distance from the origin \(\displaystyle (0,0)\) to \(\displaystyle (8,0)\), which is 8.

The area of a right triangle is half the product of the lengths of its legs \(\displaystyle a\) and \(\displaystyle b\), so, setting \(\displaystyle a = 8\) and \(\displaystyle A = 40\) and solving for \(\displaystyle b\):

\(\displaystyle \frac{1}{2} ab = A\)

\(\displaystyle \frac{1}{2} \cdot 8 \cdot b = 40\)

\(\displaystyle 4 b = 40\)

\(\displaystyle \frac{4 b}{4} = \frac{40}{4}\)

\(\displaystyle b = 10\)

Therefore, the length of the vertical leg is 10, and, since the \(\displaystyle y\)-intercept of the line containing the hypotenuse is on the positive \(\displaystyle y\)-axis, this intercept is \(\displaystyle (0, 10 )\). The slope of a line with intercepts \(\displaystyle (a, 0),(0, b)\) is 

\(\displaystyle m = - \frac{b}{a}\), 

so, setting \(\displaystyle a = 8\) and \(\displaystyle b = 10\):

\(\displaystyle m = - \frac{10}{8} = -\frac{5}{4}\)

Set \(\displaystyle m = -\frac{5}{4}\) and \(\displaystyle b = 10\) in the slope-intercept form of the equation of a line, 

\(\displaystyle y = mx+ b\);

the line has equation

\(\displaystyle y = -\frac{5}{4}x+10\)

The \(\displaystyle y\)-coordinate \(\displaystyle Y\) of the point on the line with \(\displaystyle x\)-coordinate 2 can be found using substitution; setting \(\displaystyle x= 2 , y = Y\)::

\(\displaystyle Y = -\frac{5}{4} (2)+10 = -\frac{5}{2} + 10 = -2\frac{1}{2} + 10 = 7 \frac{1}{2}\)

Mrs. Smith's 8th grade class had a quiz last week. The graph below depicts the number of questions students got incorrect on their quiz and their corresponding quiz grade. In other words, the graph in this particular question is a dot plot and the question asks to find a correlation if one exists.

Recall that a correlation is a trend seen in the data. Graphically, trends can be either:

I. Positive 

II. Negative

III. Constant

IV. No trend

For a trend to be positive the x and y variable both increase. A trend is negative when the y variable (dependent variable) decreases as the x variable (independent variable) increases. A constant trend occurs when the y variable stays the same as the x variable increases. No trend exists when the data appears to be scattered with no association between the x and y variables.

Examining the graph given it is seen that the x variable is the number of questions missed and the y variable is the overall score on the quiz. It is seen that as the number of questions missed increases, the overall score on the quiz decreases. This describes  a negative trend.

In other words, the graph depicts a negative correlation.

Given the graph of record sales, to find the fraction of records that were sold in 2004 to 2010 first identify the record sales in 2004 and the record sales in 2010.

Examining the graph,

Record sales in 2004:14 million

Record sales in 2010: 13 million

From here, to find the fraction of records sold during this time period, use the following formula.

\(\displaystyle \text{Fraction of Sales}=\frac{\text{Beginning Year Record Sales}}{\text{Ending Year Record Sales}}\)

\(\displaystyle \text{Fraction}=\frac{14}{13}\)