How to find length of a line

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ISEE Middle Level Quantitative Reasoning › How to find length of a line

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What is the length of a line with endpoints and .

Explanation

To find the length of this line, you can subtract to get . Since the y-coordinates are the same, you don't have to take any vertical direction into account. Therefore, you only look at the x-coordinates!

What is the length of a line segment with end points and ?

Explanation

The length of a line segment can be determined using the distance formula:

$D=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

$D=\sqrt{(3-6)^2+(5-1)^2}$

$D=\sqrt{-3^2+4^2}$

$D=\sqrt{9+16}=\sqrt{25}=5$

Find the slope of a line that passes through and .

The slope is undefined.

Explanation

Use the slope formula to solve:

$m = \frac{y_{2} -y_{1}}{x_{2} - x_{1}}$

Given the following points.

The slope can be calculated as follows.

$m = \frac{-1 - (-1)}{3- (-5)}$

$m = \frac{0}{8}$

Because the coordinates were the same for points A and B, this would form a horizontal line. The slope of any horizontal line is

If you stand on a giant number line at point , what is your distance from point ?

Explanation

The distance on a number line is the number of whole number places between two numbers. When you are going from a negative value to a positive one or vice versa, you must find the distance from that both points are then add them together. is spaces away from and is spaces away. To find the distance you then add your two values of and to get .

A right triangle has one leg with a length of and a hypotenuse of . What is the approximate measurement of the missing length?

$b\approx 7.4cm$

$b \approx 9.5cm$

$b\approx 8.3cm$

$b\approx 11cm$

Explanation

The Pythagorean Theorem states that

$a^{2} +b^{2} = c^{2}$

Where and are the measurements of each leg and is the hypotenuse.

$3^{2} = 3 \times3 =9$

$8^{2} = 8\times 8 = 64.$

$9 + b^{2} = 64$

$(9-9) + b^{2} = 64-9$

$b^{2} = 55$

$\sqrt{b^{2}} = \sqrt{55}$

$b\approx 7.4cm$

The point lies on a circle. What is the approximate length of the radius of the circle if the center is

$d=\sqrt{10}$

$d\approx 7$

$d=\sqrt{5}$

$d\approx5$

Explanation

Because the radius is the distance from center to any point on a circle, the distance formula is used to find the measurement of the radius.

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

$d = \sqrt{(3-2)^{2} + (-5--2)^{2}$

$1^{2} = 1\times1 = 1$

$-3^{2} = -3\times -3 = 9$

$d = \sqrt{1+9}$

$d =\sqrt{10}$

A ladder is leaning on a wall. It is ft long. The bottom of the ladder is from the base. How far up the wall is the top of the ladder?

$\approx 12.5ft$

$\approx11.5ft$

$\approx13.5ft$

$\approx 14ft$

Explanation

The Pythagorean Theorem states that:

$a^{2} +b^{2} = c^{2}$

With and representing the measurement of the legs and representing the hypotenuse.

$10^{2} +b^{2} = 16^{2}$

$100 + b^{2} = 256$

$(100-100) + b^{2} = 256-100$

$b^{2} = 156$

$\sqrt{b^{2}} = \sqrt{156}$

$b\approx 12.5ft$

If a circle has a radius of , what would be the length of the longest line drawn within that circle?

Explanation

The diameter of the circle would be the longest line that can be drawn within that circle.

Because radius is half of the diameter, the diameter is calculated by multiplying the radius of the circle by two.

If the radius is , then the diameter is $8m \times 2 = 16m$

Rec

The coordinates of on the rectangle are and . Find the length of the diagonal.

$d \approx 5.4$

$d\approx 6.4$

$d \approx 4.4$

$d\approx 7.4$

Explanation

A rectangle has two diagonals with the same length. Therefore, use the distance formula to calculate the distance.

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

$(7-5)^{2} = 2^{2} = 4$

$(-3-2)^{2} = -5^{2} = 25$

$\sqrt{4+25}$

$\sqrt{29} \approx 5.4$

$d \approx 5.4$

Line has a length of It is bisected at point , and the resulting segment is bisected again at point What is the length of line segment

Explanation

First, write each portion of the statement in mathematical terms.

$AC =\frac{1}{2} AB$

Since AB=80 we will substitute that into the equation.

$AC =\frac{1}{2} (80)$

Now that we know AC we can calculate AD as follows.

$AD = \frac{1}{2} AC$

$AD = \frac{1}{2}(40)$

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