### All ACT Math Resources

## Example Questions

### Example Question #1 : Distance Formula

Let W and Z be the points of intersection between the parabola whose graph is *y* = –*x*² – 2*x* + 3, and the line whose equation is *y* = *x* – 7. What is the length of the line segment WZ?

**Possible Answers:**

4

4√2

7

7√2

**Correct answer:**

7√2

First, set the two equations equal to one another.

–*x*² – 2*x* + 3 = *x* – 7

Rearranging gives

*x*² + 3*x* – 10 = 0

Factoring gives

(*x* + 5)(*x* – 2) = 0

The points of intersection are therefore W(–5, –12) and Z(2, –5)

Using the distance formula gives 7√2

### Example Question #2 : Distance Formula

In an *xy-*plane, what is the length of a line connecting points at (–2,–3) and (5,6)?

**Possible Answers:**

12.5

7.5

11.4

9.3

**Correct answer:**

11.4

Use the distance formula:

*D* = √((*y*_{2} – *y*_{1})^{2} + (*x*_{2} – *x*_{1})^{2})

*D* = √((6 + 3)^{2} + (5 + 2)^{2})

*D* = √((9)^{2} + (7)^{2})

*D* = √(81 + 49)

*D* = √130

*D* = 11.4

### Example Question #1 : How To Find The Length Of A Line With Distance Formula

What is the distance between points and , to the nearest tenth?

**Possible Answers:**

**Correct answer:**

The distance between points and is 6.4. Point is at . Point is at . Putting these points into the distance formula, we have .

### Example Question #4 : Distance Formula

What is the slope of the line between points and ?

**Possible Answers:**

**Correct answer:**

The slope of the line between points and is . Point is at . Point is at . Putting these points into the slope formula, we have .

### Example Question #5 : Distance Formula

What is the distance between and ?

**Possible Answers:**

**Correct answer:**

Let and and use the distance formula: . The distance formula is a specific application of the more general Pythagorean Theorem: .

### Example Question #6 : Distance Formula

What is the distance, in coordinate units, between the points and in the standard coordinate plane?

**Possible Answers:**

**Correct answer:**

The distance formula is , where = distance.

Plugging in our values, we get

### Example Question #7 : Distance Formula

What is the distance between points and ?

**Possible Answers:**

**Correct answer:**

**Solution A:**

Use the distance formula to calculate the distance between the two points:

**Solution B:**

Draw the two points on a coordinate graph and create a right triangle with sides 4 and 5. Using the Pythagorean Theorem, solve for the hypotenuse or the distance between the two points:

### Example Question #8 : Distance Formula

What is the distance between (1,5) and (6,17)?

**Possible Answers:**

**Correct answer:**

Let and

So we use the distance formula

and evaluate it using the given points:

### Example Question #9 : Distance Formula

What is the area of a square with a diagonal that has endpoints at (4, **–**1) and (2, **–**5)?

**Possible Answers:**

10

20

5

100

25

**Correct answer:**

10

First, we need to find the length of the diagonal. In order to do that, we will use the distance formula:

Now that we have the length of the diagonal, we can find the length of the side of a square. The diagonal of a square makes a 45/45/90 right triangle with the sides of the square, which we shall call s. Remember that all sides of a square are equal in length.

Because this is a 45/45/90, the length of the hypotenuse is equal to the length of the side multiplied by the square root of 2

The area of the square is equal to s^{2}, which is 10.

### Example Question #2 : How To Find The Length Of A Line With Distance Formula

Line segment has end points of and .

Line segemet has end points of and .

What is the distance between the midpoints?

**Possible Answers:**

**Correct answer:**

The midpopint is found by taking the average of each coordinate:

and

The distance formula is given by

.

Making the appropriate substitutions we get a distance of 13.