How to find length of a line - ISEE Middle Level Math

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Question

Figure NOT drawn to scale.

Evaluate .

Answer

By the Segment Addition Postulate,

$5x \div 5 = 43 \div 5$

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Question

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

Answer

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

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

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

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

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

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Question

What is the length of a line with endpoints and .

Answer

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!

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Question

Find the length of the line segment whose endpoints are and .

Answer

We can use the distance formula:

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

$=\sqrt{(-3-5)^2+(5-4)^2}$

$=\sqrt{64+1}$

$=\sqrt{65}\approx 8.06$

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Question

The point lies on a circle. What is the length of the radius of the circle if the center is located at ?

Answer

The radius is the distance from the center of the circle to anypoint on the circle. So we can use the distance formula in order to find the radius of the circle:

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

$=\sqrt{(4-3)^2+(4-8)^2}$

$=\sqrt{1+16}$

$=\sqrt{17}\approx 4.12$

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Question

The coordinates of and are and . Find the length of the diagonal of the following rectangle:

Answer

A rectangle has two diagonals with the same length. So we should find the length of . We can use the distance formula:

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

$=\sqrt{(4-7)^2+(1+1)^2}$

$=\sqrt{9+4}$

$=\sqrt{13}\approx 3.6$

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Question

If a circle has a radius of 12 inches, the biggest line that would be drawn within the circle is:

Answer

The largest line that can be drawn within a circle is the diamater. The diameter is equal to twice the radius. Given that the radius is equal to 12 inches, the largest line that could be drawn (the diameter) would be equal to 24 inches.

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Question

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

Answer

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 .

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Question

A line has endpoints and What is the distance of the line?

Answer

Use the distance formula

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

$d=\sqrt{(7-1)^{2} + (9-1)^{2}}$

$d=\sqrt{36+64}$

$d = \sqrt{100}$

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Question

One of the legs on a right triangle has a measurement of and the other leg has a measurement of What is the length of the hypotenuse?

Answer

The Pythagorean Theorem states:

$a^{2 } +b^{2} =c ^{2}$ where represents the hypotenuse and and represent the measurements of the legs of the right triangle.

$30^{2} = 30 \times30 = 900$

$40^{2} = 40 \times 40 = 1600$

$\sqrt{c^{2}} = \sqrt{2500}$

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Question

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

Answer

The Pythagorean Theorem states that

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

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

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

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

$9 + b^{2} = 64$

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

$b^{2} = 55$

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

$b\approx 7.4cm$

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Question

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

Answer

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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Question

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

Answer

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.

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

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

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

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

$d = \sqrt{1+9}$

$d =\sqrt{10}$

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Question

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

Answer

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

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

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

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

$\sqrt{4+25}$

$\sqrt{29} \approx 5.4$

$d \approx 5.4$

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Question

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

Answer

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 $8m \times 2 = 16m$

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Question

If the diameter of a circle is , then what is $\frac{1}{3}$ of the circle's radius?

Answer

Radius is $\frac{1}{2}$ of the diameter.

If the diameter is , then the radius is

$\frac{1}{2}\cdot \frac{12}{1}=\frac{12}{2}=\frac{2\cdot6}{2}=6$

Therefore

$\frac{1}{3}$ of is,

$\frac{1}{3}\cdot \frac{6}{1}=\frac{6}{3}=\frac{3\cdot 2}{3}=2cm$ .

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Question

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?

Answer

The Pythagorean Theorem states that:

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

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

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

$100 + b^{2} = 256$

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

$b^{2} = 156$

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

$b\approx 12.5ft$

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Question

What is the perimeter of a right triangle if the hypotenuse is and the measurement of one of its legs is

Answer

First, use the Pythagorean Theorem to get the measurement of the other leg.

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

$9^{2} + b^{2} =15^{2}$

$81 + b^{2} = 225$

$(81-81) + b^{2} = 225-81$

$b^{2} = 144$

To get the perimeter of the triangle, add the measurements of each of the three sides.

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Question

Find the slope of a line that passes through and .

Answer

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

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Question

Find the length of a line from the point to the point .

Answer

Find the length of a line from the point to the point .

To find this distance, we need to use distance formula (which is really similar to Pythagorean Theorem)

Distance formula is as follows

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

Where our x's and y's come from our ordered pairs.

So, let's plug and chug

$d=\sqrt{(9--12)^2+(14-36)^2}$

Simplify

$d=\sqrt{(21)^2+(-22)^2}$

$d=\sqrt{441+484}=\sqrt{925}\approx30.4$

So our answer is 30.4

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