### All SAT Math Resources

## Example Questions

### Example Question #217 : Coordinate Geometry

Give the area of the triangle on the coordinate plane that is bounded by the -axis, and the lines of the equations and .

**Possible Answers:**

None of the other choices gives the correct response.

**Correct answer:**

It is necessary to find the vertices of the triangle, which can be done by finding the three points at which two of the three lines intersect.

The intersection of the -axis - the line - and the line of the equation , is found by noting that if , then, by substitution, ; this point of intersection is at , the origin.

The intersection of the -axis and the line of the equation is the -intercept of the latter line. Since its equation is written in slope-intercept form , with the -coordinate of the -intercept, this intercept is .

The intersection of the lines with equations and can be found using the substitution method, setting in the latter equation and solving for :

Since , , making the point of intersection.

The lines in question are graphed below, and the triangle they bound is shaded:

If we take the vertical side as the base, its length is seen to be 3; the height is the horizontal distance to the opposite vertex, which is its -coordinate . The area is half the product of the two, or

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### Example Question #218 : Coordinate Geometry

Give the area of the triangle on the coordinate plane that is bounded by the lines of the equations , and .

**Possible Answers:**

**Correct answer:**

It is necessary to find the coordinates of the vertices of the triangle, each of which is the intersection of two of the three lines.

The intersection of the lines of the equations and can easily be found by noting that since , by substitution, , making the point of intersection .

The intersection of the lines of the equations and can be found by substituting for in the latter equation and evaluating :

The point of intersection is .

The intersection of the lines of the equations and can be found by substituting for in the latter equation and solving for :

, so , and this point of intersection is .

The lines in question are graphed below, and the triangle they bound is shaded:

We can take the vertical side as the base of the triangle; its length is the difference of the -coordinates:

The height is the horizontal distance from this side to the opposite side, which is the difference of the -coordinates:

The area is half their product:

### Example Question #221 : Coordinate Geometry

Give the area of the triangle on the coordinate plane that is bounded by the axes and the line of the equation .

**Possible Answers:**

**Correct answer:**

It is necessary to find the vertices of the triangle, each of which is a point at which two of the three lines intersect.

The two axes intersect at the origin, making this one vertex.

The other two points of intersection are the intercepts of the line of equation . Since this equation is in slope-intercept form , where is the -coordinate of the -intercept, then , and the -intercept is the point . The -coordinate of the -intercept, , can be found by setting and solving for :

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The three vertices are located at

The line in question is shown below, with the bounded triangle shaded in:

The lengths of its legs are equal to the absolute values of the nonzero coordinates of its two intercepts - . The area of this right triangle is half their product:

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### Example Question #21 : How To Find X Or Y Intercept

What is the value of the -intercept for the line given below?

**Possible Answers:**

**Correct answer:**

The x-intercept is where the line crosses the x-axis. In other words,

This gives:

Subtracting 98 from both sides gives:

Dividing both sides by 14 gives the final answer:

### Example Question #22 : How To Find X Or Y Intercept

Give the area of the triangle on the coordinate plane that is bounded by the -axis, and the lines of the equations and

**Possible Answers:**

**Correct answer:**

It is necessary to find the vertices of the triangle, each of which is a point at which two of the three lines intersect.

The intersection of the -axis - the line - and the line of the equation , is found by noting that if , then, by substitution, ; this point of intersection is at , the origin.

The intersection of the -axis and the line of the equation is found similarly:

This intersection point is at .

The intersection of the lines with equations and can be found using the substitution method, setting in the latter equation and solving for :

Since , , making the point of intersection.

The vertices are at .

The lines in question are graphed below, and the triangle they bound is shaded:

If we take the horizontal side as the base, its length is seen to be the -coordinate of the -intercept, ; its (vertical) height is the -coordinate of the opposite vertex, . The area is half the product of the two, or

### Example Question #611 : Geometry

Given the line , what is the sum of the -intercept and the -intercept?

**Possible Answers:**

**Correct answer:**

Intercepts occur when a line crosses the -axis or the -axis. When the line crosses the -axis, then and . When the line crosses the -axis, then and . The intercept points are and . So the -intercept is and the intercept is and the sum is .

### Example Question #612 : Geometry

Where does the line given by intercept the -axis?

**Possible Answers:**

**Correct answer:**

First, put in slope-intercept form.

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To find the -intercept, set and solve for .