Distance Formula

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ACT Math › Distance Formula

Questions 1 - 10

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 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 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 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 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

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

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