# GRE Subject Test: Math : Distance & Midpoint Formulas

## Example Questions

### Example Question #11 : Find The Distance Between A Point And A Line

Find the distance between and the point

Explanation:

To find the distance, use the formula where the point is and the line is

First, we'll re-write the equation  in this form to identify , , and :

subtract from and add  to both sides

multiply both sides by

Now we see that . Plugging these plus  into the formula, we get:

### Example Question #1 : Find The Distance Between Two Parallel Lines

Find the distance  between the two lines.

Explanation:

Since the slope of the two lines are equivalent, we know that the lines are parallel. Therefore, they are separated by a constant distance. We can then find the distance between the two lines by using the formula for the distance from a point to a nonvertical line:

First, we need to take one of the line and convert it to standard form.

where

Now we can substitute A, B, and C into our distance equation along with a point, , from the other line. We can pick any point we want, as long as it is on line . Just plug in a number for x, and solve for y. I will use the y-intercept, where x = 0, because it is easy to calculate:

Now we have a point, , that is on the line . So let's plug our values for :

### Example Question #1 : Find The Distance Between Two Parallel Lines

Find the distance between and

Explanation:

To find the distance, choose any point on one of the lines. Plugging in 2 into the first equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

now we see that

We can plug the coefficients and the point into the formula

where represents the point.

### Example Question #1 : Find The Distance Between Two Parallel Lines

Find the distance between and

Explanation:

To find the distance, choose any point on one of the lines. Plugging in  into the second equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

multiply both sides by

now we see that

We can plug the coefficients and the point into the formula

where represents the point.

### Example Question #4 : Find The Distance Between Two Parallel Lines

How far apart are the lines and ?

Explanation:

To find the distance, choose any point on one of the lines. Plugging in  into the first equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

multiply both sides by

now we see that

We can plug the coefficients and the point into the formula

where represents the point.

### Example Question #5 : Find The Distance Between Two Parallel Lines

Find the distance between and

Explanation:

To find the distance, choose any point on one of the lines. Plugging in  into the second equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

multiply both sides by

now we see that

We can plug the coefficients and the point into the formula

where represents the point.

### Example Question #6 : Find The Distance Between Two Parallel Lines

Find the distance between and

Explanation:

To find the distance, choose any point on one of the lines. Plugging in  into the first equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

multiply both sides by

now we see that

We can plug the coefficients and the point into the formula

where represents the point.

### Example Question #2 : Find The Distance Between Two Parallel Lines

Find the distance between the lines and

Explanation:

To find the distance, choose any point on one of the lines. Plugging in into the first equation can generate our first point:

this gives us the point

We can find the distance between this point and the other line by putting the second line into the form :

subtract the whole right side from both sides

multiply both sides by

now we see that

We can plug the coefficients and the point into the formula

where represents the point.

### Example Question #11 : Coordinate Geometry

Find the distance between the points  and .

None of the Above

Explanation:

Step 1: Let's define the distance formula. The distance between two sets of coordinates can be found by using the equation:

In the equation, d is the distance. Also,  and  are the coordinate points.

, .

Step 2: Plug in the values for the missing variables into the equation:

Step 3: Simplify the inside of the square root. Remember that two minus signs next to each other will change to a plus sign.

Step 4: Add up the numbers in the parentheses:

Step 5: Evaluate the exponents:

Step 6: Add the numbers under the square root.

Step 7: Simplify the number inside the square root as much as possible.

Let's divide by 4:

. We cannot break down 145 into another perfect square, so it has to go back into the radical. The square root of 4 is 2, and this will go on the outside.

### Example Question #12 : Coordinate Geometry

Find the distance between the points  and

Explanation:

Step 1: The distance formula is defined as:

.

Step 2. Identify what  and  are.

Step 3: Substitute each value for its place in the distance formula.

We will get this:

Step 4: Simplify the inside of step 3.

Step 5: Simplify the parentheses:

Step 6: Evaluate each exponent:

Step 7: Reduce  to lowest terms:

Divide  by :

Step 8: Rewrite

Replace  with :

The simplified answer to the question is