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ISEE Upper Level Quantitative : Lines

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

Example Question #81 : Plane Geometry

What is the slope of a line that passes through points (4.5,2.5) and (3.5, 6.5) ?

Possible Answers:

$-\frac{1}{4}$

$\frac{1}{4}$

Correct answer:

Explanation:

The equation for solving for the slope of a line is $m=\frac{y2-y1}{x2-x1}$ .

Thus, if (x1,y1)=(4.5, 2.5) and (x2,y2)=(3.5,6.5) , then:

$m=\frac{6.5-2.5}{3.5-4.5}=\frac{4}{-1}=-4$

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Example Question #11 : Lines

What is the slope of a line that passes through points (4,12) and (8, 20) ?

Possible Answers:

0.5

2.5

Correct answer:

Explanation:

The equation for solving for the slope of a line is $m=\frac{y2-y1}{x2-x1}$ .

Thus, if (x1,y1)=(4,12) and (x2,y2)=(8,20) , then:

$m=\frac{20-12}{8-4}=\frac{8}{4}=2$

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Example Question #92 : Plane Geometry

Tangents 1

Figure NOT drawn to scale

In the above figure, $\overline{TN}$ and $\overline{TO}$ are tangent segments. The ratio of the length of $\overarc{NPO}$ to that of $\overarc{N O}$ is 5 to 3. Which is the greater quantity?

(a) $m \angle NTO$

(b) $30 ^{\circ }$

Possible Answers:

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

For the sake of simplicity, let us assume that the lengths of $\overarc{NPO}$ and $\overarc{N O}$ are 5 and 3; this reasoning depends only on their ratio and not their actual length. The circumference of the circle is the sum of the lengths, which is 8, so $\overarc{NPO}$ and $\overarc{N O}$ comprise $\frac{5}{8}$ and $\frac{3}{8}$ of the circle, respectively. Therefore,

$m \overarc{NPO} = \frac{5}{8} \cdot 360 ^{\circ } = 225 ^{\circ }$ ; and

$m \overarc{N O} = \frac{3}{8} \cdot 360 ^{\circ } = 135 ^{\circ }$ .

If two tangents are drawn to a circle, the measure of the angle they form is half the difference of the measures of the arcs they intercept, so

$m \angle NTO = \frac{1}{2}(m \overarc{NPO} - m \overarc{N O} ) = \frac{1}{2}(225 ^{\circ }- 135^{\circ } ) = \frac{1}{2}(90^{\circ } ) = 45^{\circ }$

This is greater than $30 ^{\circ }$ .

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Example Question #11 : Lines

Which is the greater quantity?

(a) The length of the line segment connecting (0,0) and (a,b)

(b) The length of the line segment connecting (a,0) and (0,b)

Possible Answers:

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Correct answer:

(a) and (b) are equal.

Explanation:

(a) The length of the line segment connecting (0,0) and (a,b) is

$\sqrt{(a -0)^{2}+(b -0)^{2} } = \sqrt{a^{2}+b^{2} }$ .

(b) The length of the line segment connecting (a,0) and (0,b) is

$\sqrt{(a -0)^{2}+(0-b)^{2} } = \sqrt{a^{2}+\left ( - b \right ) ^{2} }= \sqrt{a^{2}+b^{2} }$ .

The segments have equal length.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Lines

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #81 : Plane Geometry

Example Question #11 : Lines

Example Question #92 : Plane Geometry

Example Question #11 : Lines

All ISEE Upper Level Quantitative Resources

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