### All SAT Math Resources

## Example Questions

### Example Question #261 : Coordinate Geometry

On the coordinate plane, a triangle has its vertices at the points with coordinates

, , and . Give the coordinates of the center of the circle that circumscribes this triangle.

**Possible Answers:**

**Correct answer:**

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 and can be found using the slope formula, setting :

The slope of the line that connects and can be found similarly, setting :

The product of their slopes is , which indicates perpendicularity between the sides.

This makes the triangle right, and the side with endpoints and 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, .

### Example Question #1 : How To Graph A Point

On the coordinate plane, a triangle has its vertices at the points with coordinates , , and . Give the coordinates of the center of the circle that circumscribes this triangle.

**Possible Answers:**

**Correct answer:**

The referenced figure is below.

The triangle formed is a right triangle whose hypotenuse is the segment with the endpoints and . The center of the circle that circumscribes a right triangle is the midpoint of its hypotenuse, so the midpoint formula

can be applied, setting :

The midpoint of the hypotenuse, and, consequently, the center of the circumscribed circle, is the point with coordinates .

### Example Question #2 : How To Graph A Point

On the coordinate plane, , , and are the points with coordinates , , and , respectively. Lines , , and are the perpendicular bisectors of , , and , respectively.

and intersect at a point ; and intersect at a point ; and intersect at a point .

Which of these statements is true of , , and ?

**Possible Answers:**

, , and are distinct and are the vertices of an equilateral triangle.

, , and are distinct and collinear.

, , and are distinct and are the vertices of a triangle similar to .

and are the same point; is a different point.

, , and are the same point.

**Correct answer:**

, , and are the same point.

Another way of viewing this problem is to note that the three given vertices form a triangle whose sides' perpendicular bisectors intersect at the points , , and . However, the three perpendicular bisectors of the sides of *any* triangle always intersect at a common point. The correct response is that , , and are the same point.

### Example Question #1 : How To Graph A Point

Figure NOT drawn to scale.

On the coordinate axes shown above, the shaded triangle has area 40.

Evaluate .

**Possible Answers:**

**Correct answer:**

The length of the horizontal leg of the triangle is the distance from the origin to , which is 8.

The area of a right triangle is half the product of the lengths of its legs and , so, setting and and solving for :

Therefore, the length of the vertical leg is 10, and, since the -intercept of the line containing the hypotenuse is on the positive -axis, this intercept is . The slope of a line with intercepts is

,

so, setting and :

Set and in the slope-intercept form of the equation of a line,

;

the line has equation

The -coordinate of the point on the line with -coordinate 2 can be found using substitution; setting ::

### Example Question #663 : Geometry

Mrs. Smith's 8th grade class has a weekly quiz. The graph below depicts the number of questions students got incorrect on their quiz and their corresponding quiz grade. Examining the graph, what type of correlation if any, exists?

**Possible Answers:**

More information is needed.

The graph depicts a positive correlation.

The graph depicts a negative correlation.

The graph depicts a constant correlation.

The graph depicts no correlation.

**Correct answer:**

The graph depicts a negative correlation.

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.

### Example Question #2 : How To Graph A Point

Given the graph of record sales, what fraction of records were sold in 2004 to 2010?

**Possible Answers:**

**Correct answer:**

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.