How to find length of a line - ISEE Lower Level Quantitative Reasoning

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Question

On a number line, what is the length of a line that stretches from $\dpi{100} -3$ to $\dpi{100} 13$ ?

Answer

$\dpi{100} -3$ is $\dpi{100} 3$ units to the left of zero on the number line. $\dpi{100} 13$ is $\dpi{100} 13$ units to the right of zero on the number line.

The length of the line would then be $\dpi{100} 13+ 3=16$ .

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Question

If the radius of a circle is equal to $\frac{1}{4}x$ , then what is the value of the diameter?

Answer

The length of a dimater is twice the length of the radius. Given that the length of the radius is $\frac{1}{4}x$ , the dimater will be equal to $\frac{1}{4}x\cdot2=\frac{2}{4}x=\frac{1}{2}x$ .

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Question

The length of is and the length of is .

What is the length of in terms of and ?

Answer

Using the segment addition postulate, we know that . We can substitute and for and and solve for .

Substitute:

Subtract y from both sides:

Simplify:

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Question

The length of BC is x and the length of CE is y.

What is the length of segment ?

Answer

Segments and make up segment . Using the segment addition postulate, add the lengths of and to find the length of .

Substitute and :

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Question

The length of is and the length of is .

What is the length of segment ?

Answer

Using the segment addition postulate, we know that . We can substitute and for and and solve for .

Substitute:

Subtract y from both sides:

Simplify:

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Question

The length of is and the length of is .

What is the length of segment in terms of and ?

Answer

Segments and make up segment . Using the segment addition postulate, add the lengths of and to find the length of .

Substitute and :

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Question

What is the distance between the points and ?

Answer

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

Let $P_{1}= (-2, 1)$ and $P_{2}=(10, -4)$ .

Plug these two points into the distance formula:

$d = \sqrt{(10-(-2))^{2}+(-4-1)^{2}}=\sqrt{(12)^{2}+(-5)^{2}}=\sqrt{144+25}=\sqrt{169}=13$

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Question

If a line with a length of 5 starts at the point , which of the following is NOT a possible end point?

Answer

Let $P_{1}=(1,1)$ , and let $P_{2}$ be the end point we are looking for.

Recall the distance formula:

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

Squaring the distance formula gives $d^{2} = (x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}$ .

Plug $P_{1}=(1,1)$ this squared distance formula and solve for $P_{2}$ :

$25 = (x_{2}-1)^{2}+(y_{2}-1)^{2}$

Substitute each response into this equation to see which one is false. The only response that doesn't work is .

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Question

What is the distance between and ?

Answer

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

Let $P_{1}= (-2, 1)$ and $P_{2}=(1, 5)$ .

Plug these points into the distance formula:

$d = \sqrt{(1-(-2))^{2}+(5-1)^{2}}=\sqrt{(3)^{2}+(4)^{2}}=\sqrt{9+16}=\sqrt{25}=5$

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Question

If a line that is 13 inches in length starts at the point , which of the following is NOT a possible end point?

Answer

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

Square both sides:

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

Now, let $P_{1}=(1,2)$ , and let $P_{2}$ be the end point we are looking for.

Plug $P_{1}=(1,2)$ into the squared distance formula and solve for $P_{2}$ :

$169 = (x_{2}-1)^{2}+(y_{2}-2)^{2}$

Alternatively, you can plug in the answer choices and see which point makes the above equation untrue. The point matches this description and is therefore the correct answer.

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Question

What is the distance between and ?

Answer

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

Let $P_{1}=(4,-5)$ and $P_{2}=(-2,3)$ :

$d=\sqrt{(-2-4)^{2} + (3-(-5)^{2}}=\sqrt{(-6)^{2} + (8)^{2}}=\sqrt{36 + 64}=\sqrt{100}=10$

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Question

What is the distance between and ?

Answer

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

Let $P_{1}=(1,-3)$ and $P_{2}=(13,-8)$ :

$d=\sqrt{(13-1)^{2} + (-8-(-3)^{2}}=\sqrt{(12)^{2} + (-5)^{2}}=\sqrt{144 + 25}=\sqrt{169}=13$

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Question

What is the distance between and ?

Answer

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

Let $P_{1}=(-7,17)$ and $P_{2}=(2,5)$ :

$d=\sqrt{(2-(-7))^{2} + (5-17)^{2}}=\sqrt{(9)^{2} + (-12)^{2}}=\sqrt{81 + 144}=\sqrt{225}=15$

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Question

How long is a line segment on the number line if it goes from to ?

Answer

To find the length of a line on the number line when one point is negative and the other is positive, you must the the distances of both from zero. is spaces from zero where is the other point is just . So the distance is .

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Question

What is the distance between and

Answer

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

Let $P_{1}=(-7,17)$ and $P_{2}=(2,5)$ :

$d=\sqrt{(2-(-7))^{2} + (5-17)^{2}}=\sqrt{(9)^{2} + (-12)^{2}}=\sqrt{81 + 144}=\sqrt{225}=15$

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Question

On a number line, what is the length of a line that stretches from $\dpi{100} -3$ to $\dpi{100} 13$ ?

Answer

$\dpi{100} -3$ is $\dpi{100} 3$ units to the left of zero on the number line. $\dpi{100} 13$ is $\dpi{100} 13$ units to the right of zero on the number line.

The length of the line would then be $\dpi{100} 13+ 3=16$ .

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Question

If the radius of a circle is equal to $\frac{1}{4}x$ , then what is the value of the diameter?

Answer

The length of a dimater is twice the length of the radius. Given that the length of the radius is $\frac{1}{4}x$ , the dimater will be equal to $\frac{1}{4}x\cdot2=\frac{2}{4}x=\frac{1}{2}x$ .

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Question

The length of is and the length of is .

What is the length of in terms of and ?

Answer

Using the segment addition postulate, we know that . We can substitute and for and and solve for .

Substitute:

Subtract y from both sides:

Simplify:

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Question

The length of BC is x and the length of CE is y.

What is the length of segment ?

Answer

Segments and make up segment . Using the segment addition postulate, add the lengths of and to find the length of .

Substitute and :

Compare your answer with the correct one above

Question

The length of is and the length of is .

What is the length of segment ?

Answer

Using the segment addition postulate, we know that . We can substitute and for and and solve for .

Substitute:

Subtract y from both sides:

Simplify:

Compare your answer with the correct one above

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