ACT Math : How to find the length of a line with distance formula

Example Questions

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

Let W and Z be the points of intersection between the parabola whose graph is y = –x² – 2x + 3, and the line whose equation is y = x – 7. What is the length of the line segment WZ?

4

7√2

4√2

7

7√2

Explanation:

First, set the two equations equal to one another.

x² – 2x + 3 = x – 7

Rearranging gives

x² + 3x – 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 #1 : How To Find The Length Of A Line With Distance Formula

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

7.5

12.5

11.4

9.3

11.4

Explanation:

Use the distance formula:

D = √((y2 – y1)2 + (x2 – x1)2)

D = √((6 + 3)2 + (5 + 2)2)

D = √((9)2 + (7)2)

D = √(81 + 49)

D = √130

D = 11.4

Example Question #3 : How To Find The Length Of A Line With Distance Formula

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

Explanation:

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

What is the slope of the line between points  and ?

Explanation:

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

What is the distance between  and ?

Explanation:

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

Example Question #1 : Distance Formula

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

Explanation:

The distance formula is , where  = distance.

Plugging in our values, we get

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

What is the distance between points  and ?

Explanation:

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

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

Explanation:

Let and

So we use the distance formula

and evaluate it using the given points:

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

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

10

5

25

20

100

10

Explanation:

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 s2, which is 10.

Example Question #121 : Algebra

Line segment  has end points of  and .

Line segemet  has end points of  and .

What is the distance between the midpoints?

Explanation:

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.

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