DSQ: Understanding intersecting lines - GMAT Quantitative

When we have a linear equation and a quadratic equation there are only so many times they can intersect. They can intersect 0 times, once, or twice.

I) Gives us the slope of one equation.

II) Gives us the vertex of our quadratic equation.

If you draw a picture, it should be apparent that we don't have enough information to know exactly how many times they intersect. Our quadratic could be facing up or down, and our linear equation could go straight through both arms, or it could miss it entirely. Therfore, neither statement is sufficient.

Statement 1: and

The line is a horizontal line on the x-axis. The line is a vertical line graphed along the y-axis. The lines will create perpendicular angles, which are all 90 degrees.

Statement 2: and

These two functions are in form, which allows us to determine the slopes of these functions. The slopes are 2 and negative half, which are both the negative reciprocal to each other. The property of negative reciprocal slopes state that these two lines are also perpendicular to each other.

Therefore:

Statement 1): Two angles are acute, and two angles are obtuse.

This statement is not necessarily true. Two intersecting lines may also be perpendicular to each other, which means that all four angles are 90 degrees.

There is not enough information to justify this statement.

Statement 2: Any two non-perpendicular intersecting lines with known equations.

This is a tricky statement.

When two functions meet, they must have an intersecting point . Both functions can be set equal to each other to determine that intersecting point.

Draw an imaginary line where the line is perpendicular to the first function and passes through the second function at some known arbitrary point . Point will need to be determined.

The equation of the third function can be determined since imaginary line intersects equation at , and is also perpendicular to . The slope of can be determined since it's the negative reciprocal of the slope of .

After the equation has been determined by using point and the slope of , the point can also be determined by setting the functions equal to each other.

Once the points have been determined, the distance formula may be used to determine the lengths from , , and .

The Law of Sines can then be used to determine the interior angles of the triangle bounded by . Knowing one angle at the intersection of is sufficient to solve for all four angles by supplementary and opposite angle rules.

Therefore:

With just statement 1, we know , so we could determine the measure of angle y, but there is no definitive relationship between z and x nor y and x, because we don't know if lines p and q are parallel.

With just statement 2,we know , so we could determine the measure of angle z, but there is no definitive relationship between y and x nor z and x, because we don't know if lines p and q are parallel.

Even if we have the information from both statements 1 and 2, we still do not know if lines p and q are parallel, therefore there is no difinitive relationship between angle y and angle x nor angle z and angle x.

With just statement 1, there is no definitive relationship between angle y and angle x.

With just statement 2, there is a difinitive relationship between angle y and angle x, but we don't know the measure of angle y.

If you have the information from both statements 1 and 2, you can determine the measure of , so .

Fortunately, this is a data sufficiency question, so you don't have to actually do the math, you just have to know that you have all the information to do the math.

Although the question itself tells us that lines p and q are parallel, the iinformation in statement 1 is insufficient to determine a definitive value for either y or z.

With the information from the question that lines p and q are parallel, and the added information from statement 2 that , using the rules of supplementary angles and alternate interior angles, we can determine the value of x.

If we know both statements, then we know that the segment can be either or , since each has endpoint and each includes ; we can not eliminate either, however.

When we have a linear equation and a quadratic equation there are only so many times they can intersect. They can intersect 0 times, once, or twice.

I) Gives us the slope of one equation.

II) Gives us the vertex of our quadratic equation.

If you draw a picture, it should be apparent that we don't have enough information to know exactly how many times they intersect. Our quadratic could be facing up or down, and our linear equation could go straight through both arms, or it could miss it entirely. Therfore, neither statement is sufficient.

Statement 1: and

The line is a horizontal line on the x-axis. The line is a vertical line graphed along the y-axis. The lines will create perpendicular angles, which are all 90 degrees.

Statement 2: and

These two functions are in form, which allows us to determine the slopes of these functions. The slopes are 2 and negative half, which are both the negative reciprocal to each other. The property of negative reciprocal slopes state that these two lines are also perpendicular to each other.

Therefore:

Statement 1): Two angles are acute, and two angles are obtuse.

This statement is not necessarily true. Two intersecting lines may also be perpendicular to each other, which means that all four angles are 90 degrees.

There is not enough information to justify this statement.

Statement 2: Any two non-perpendicular intersecting lines with known equations.

This is a tricky statement.

When two functions meet, they must have an intersecting point . Both functions can be set equal to each other to determine that intersecting point.

Draw an imaginary line where the line is perpendicular to the first function and passes through the second function at some known arbitrary point . Point will need to be determined.

The equation of the third function can be determined since imaginary line intersects equation at , and is also perpendicular to . The slope of can be determined since it's the negative reciprocal of the slope of .

After the equation has been determined by using point and the slope of , the point can also be determined by setting the functions equal to each other.

Once the points have been determined, the distance formula may be used to determine the lengths from , , and .

The Law of Sines can then be used to determine the interior angles of the triangle bounded by . Knowing one angle at the intersection of is sufficient to solve for all four angles by supplementary and opposite angle rules.

Therefore:

With just statement 1, we know , so we could determine the measure of angle y, but there is no definitive relationship between z and x nor y and x, because we don't know if lines p and q are parallel.

With just statement 2,we know , so we could determine the measure of angle z, but there is no definitive relationship between y and x nor z and x, because we don't know if lines p and q are parallel.

Even if we have the information from both statements 1 and 2, we still do not know if lines p and q are parallel, therefore there is no difinitive relationship between angle y and angle x nor angle z and angle x.

With just statement 1, there is no definitive relationship between angle y and angle x.

With just statement 2, there is a difinitive relationship between angle y and angle x, but we don't know the measure of angle y.

If you have the information from both statements 1 and 2, you can determine the measure of , so .

Fortunately, this is a data sufficiency question, so you don't have to actually do the math, you just have to know that you have all the information to do the math.

Although the question itself tells us that lines p and q are parallel, the iinformation in statement 1 is insufficient to determine a definitive value for either y or z.

With the information from the question that lines p and q are parallel, and the added information from statement 2 that , using the rules of supplementary angles and alternate interior angles, we can determine the value of x.

If we know both statements, then we know that the segment can be either or , since each has endpoint and each includes ; we can not eliminate either, however.

When we have a linear equation and a quadratic equation there are only so many times they can intersect. They can intersect 0 times, once, or twice.

I) Gives us the slope of one equation.

II) Gives us the vertex of our quadratic equation.

If you draw a picture, it should be apparent that we don't have enough information to know exactly how many times they intersect. Our quadratic could be facing up or down, and our linear equation could go straight through both arms, or it could miss it entirely. Therfore, neither statement is sufficient.

Statement 1: and

The line is a horizontal line on the x-axis. The line is a vertical line graphed along the y-axis. The lines will create perpendicular angles, which are all 90 degrees.

Statement 2: and

These two functions are in form, which allows us to determine the slopes of these functions. The slopes are 2 and negative half, which are both the negative reciprocal to each other. The property of negative reciprocal slopes state that these two lines are also perpendicular to each other.

Therefore:

Statement 1): Two angles are acute, and two angles are obtuse.

This statement is not necessarily true. Two intersecting lines may also be perpendicular to each other, which means that all four angles are 90 degrees.

There is not enough information to justify this statement.

Statement 2: Any two non-perpendicular intersecting lines with known equations.

This is a tricky statement.

When two functions meet, they must have an intersecting point . Both functions can be set equal to each other to determine that intersecting point.

Draw an imaginary line where the line is perpendicular to the first function and passes through the second function at some known arbitrary point . Point will need to be determined.

The equation of the third function can be determined since imaginary line intersects equation at , and is also perpendicular to . The slope of can be determined since it's the negative reciprocal of the slope of .

After the equation has been determined by using point and the slope of , the point can also be determined by setting the functions equal to each other.

Once the points have been determined, the distance formula may be used to determine the lengths from , , and .

The Law of Sines can then be used to determine the interior angles of the triangle bounded by . Knowing one angle at the intersection of is sufficient to solve for all four angles by supplementary and opposite angle rules.

Therefore:

With just statement 1, we know , so we could determine the measure of angle y, but there is no definitive relationship between z and x nor y and x, because we don't know if lines p and q are parallel.

With just statement 2,we know , so we could determine the measure of angle z, but there is no definitive relationship between y and x nor z and x, because we don't know if lines p and q are parallel.

Even if we have the information from both statements 1 and 2, we still do not know if lines p and q are parallel, therefore there is no difinitive relationship between angle y and angle x nor angle z and angle x.

With just statement 1, there is no definitive relationship between angle y and angle x.

With just statement 2, there is a difinitive relationship between angle y and angle x, but we don't know the measure of angle y.

If you have the information from both statements 1 and 2, you can determine the measure of , so .

Fortunately, this is a data sufficiency question, so you don't have to actually do the math, you just have to know that you have all the information to do the math.