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

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

Explanation

Let $P_{1}=(1,5)$ and $P_{2}=(6,17)$

So we use the distance formula $d =\sqrt{(x_{2} - x_{1})^2+(y_{2} - y_{1})^2}$

and evaluate it using the given points:

$d=\sqrt{(6-1)^2+(17 - 5)^2}= \sqrt{(5)^2+(12)^2}=13$

What is the distance of the line

Between and ?

Round to the nearest hundredth.

Explanation

What is the distance of the line

Between and ?

To calculate this, you need to know the points for these two values. To find these, substitute in for the two values of given:

Likewise, do the same for :

Now, this means that you have two points:

and

The distance formulat between two points is:

$\sqrt{(x_1-x_0)^2+(y_1-y_0)^2}$

For our data, this is:

$\sqrt{(7-4)^2+(10-12)^2}=\sqrt{(3)^2+(-2)^2}=\sqrt{9+4}=\sqrt{13}$

This is:

or approximately

What is the distance between the x and y intercepts of:

Round to the nearest hundredth.

Explanation

In order to find the distance between these two points, we first need the points! To find them, substitute in for x and y. Thus, for each you get:

Thus, the two points are:

and

The distance formulat between two points is:

$\sqrt{(x_1-x_0)^2+(y_1-y_0)^2}$

For your data, this is very simply:

$\sqrt{(12)^2+(24)^2}=\sqrt{576 + 144} = \sqrt{720}$

$\sqrt{720}=26.83281572999748....$ or

What is the slope of the line between points $\dpi{100} \small A$ and $\dpi{100} \small B$ ?

$\frac{5}{4}$

$\frac{-5}{4}$

$\frac{5}{2}$

$5$

$-4$

Explanation

The slope of the line between points $\dpi{100} \small A$ and $\dpi{100} \small B$ is $\frac{5}{4}$ . Point $\dpi{100} \small A$ is at $\dpi{100} \small (-2,-3)$ . Point $\dpi{100} \small B$ is at $\dpi{100} \small (2,2)$ . Putting these points into the slope formula, we have $\frac{-3-2}{-2-2}=\frac{-5}{-4}=\frac{5}{4}$ .

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

100

Explanation

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

Actmath_29_372_q6_1

Actmath_29_372_q6_2

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

Actmath_29_372_q6_3

Actmath_29_372_q6_4_copy

The area of the square is equal to s2, which is 10.

What is the distance, in coordinate units, between the points $(-2,6)$ and $(5,-2)$ in the standard $(x,y)$ coordinate plane?

$\sqrt{113}$

$\sqrt{15}$

$15$

$113$

$\sqrt{7}$

Explanation

The distance formula is $\sqrt{((x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2})}=d$ , where $d$ = distance.

Plugging in our values, we get

$d=\sqrt{((5-(-2))^{2}+(6-(-2))^{2}}=\sqrt{7^{2}+8^{2}}=\sqrt{49+64}=\sqrt{113}$

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

11.4

12.5

7.5

9.3

Explanation

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

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:

$P_{mid} = (\frac{x_{1}+x_{2} }{2},\frac{y_{1}+y_{2} }{2})$

$AB\ mid = (2,1)$ and $CD\ mid = \left ( 7,13 \right )$

The distance formula is given by

$d = \sqrt{(x_{2} - x_{1})^2+(y_{2} - y_{1})^2}$ .

Making the appropriate substitutions we get a distance of 13.

Two towns—Town A and Town B—are represented by points on a map overlaid with a standard x- and y-coordinate plane. Town A and Town B are represented by points and , respectively. If each unit on the map represents an actual distance of 20 miles, which of the following is closest to the actual distance, in miles, between these two towns?

169

260

261

Explanation

In order to find the actual distance between these two towns, we must find the distance between these two points. We can find the distance between the points by using the distance formula, where the variable, , represents the distance:

$d = \sqrt{(x_{2}-x_{1})^2+ (y_{2}-y_{1})^2}$

The variables $x_{2}$ and $x_{1}$ represent the x-values at each point and $y_{2}$ and $y_{1}$ represent the y-values at each point. We can calculate the distance by substituting the x- and y-coordinates of each point into the distance formula.

$d = \sqrt{(-2-3)^2+ (15-3)^2}$

$d= \sqrt{169}$

Since each unit on the map represents an actual distance of miles, the actual distance between the two cities can be calculated using the the following operation:

$13 \times20 $ miles = 260 miles.$

What is the distance between $\left ( 1,-2 \right )$ and $\left ( 6,10 \right )$ ?

Explanation

Let $P_{1}=(1,-2)$ and $P_{2}=(6,10)$ and use the distance formula: $d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$ . The distance formula is a specific application of the more general Pythagorean Theorem: $a^{2} + b^{2} = c^{2}$ .

How to find the length of a line with distance formula

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