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ISEE Lower Level Math : Lines

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Example Questions

Example Question #1 : Lines

There is a four sided figure in which none of the lines run parallel to each other. Which of the following could be the appropriate term to describe the figure?

Possible Answers:

None of these

Parallelogram

Rectangle

Trapezoid

Quadrilateral

Correct answer:

Quadrilateral

Explanation:

A key characteristic of a rectangle, parallelogram, and trapezoid is the fact that they each have at least one pair of lines that run parallel to each other. The only option that has lines that may not run parallel to each other is the quadrilateral, which must simply have four sides but has no specifications about parallelism.

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Example Question #1 : Lines

If a diagonal is drawn from one corner of a rectangle to the opposite corner, what 2 shapes result?

Possible Answers:

$2\ \textup{triangles}$ $\displaystyle 2\ \textup{triangles}$

$2\ \textup{rectangles}$ $\displaystyle 2\ \textup{rectangles}$

$2\ \textup{quadrilaterals}$ $\displaystyle 2\ \textup{quadrilaterals}$

$2\ \textup{squares}$ $\displaystyle 2\ \textup{squares}$

Correct answer:

$2\ \textup{triangles}$ $\displaystyle 2\ \textup{triangles}$

Explanation:

While drawing a line across a rectangle (so that it bisects 2 sides) can result in 2 quadrilaterals, squares, or rectangles, a line drawn from one corner to the furthest corner results in two triangles. Therefore, the correct answer choice is 2 triangles.

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Example Question #1 : Lines

How many lines of symmetry are there in a square?

Possible Answers:

$\displaystyle 5$

$\displaystyle 2$

$\displaystyle 4$

$\displaystyle 3$

$\displaystyle 1$

Correct answer:

$\displaystyle 4$

Explanation:

A line of symmetry is a line that divides a polygon in half and each half is a mirror image of the other. In other words, you can fold the polygon over the symmetry line and each half matches up perfectly.

For a square there are four lines of symmetry. Two are from the diagonals of the square and two are from connecting the midpoints of the opposite sides.

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Example Question #391 : Geometry

What is the distance between (2,5) $\displaystyle (2,5)$ and $\displaystyle (-7, 17)$ ?

Possible Answers:

$\displaystyle 13$

$\displaystyle 14$

$\displaystyle 12$

$\displaystyle 16$

$\displaystyle 15$

Correct answer:

$\displaystyle 15$

Explanation:

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

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

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

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Example Question #1331 : Isee Lower Level (Grades 5 6) Mathematics Achievement

What is the distance between (-2, 3) $\displaystyle (-2, 3)$ and (4, -5) $\displaystyle (4, -5)$ ?

Possible Answers:

$\displaystyle 12$

$\displaystyle 8$

$\displaystyle 10$

$\displaystyle 14$

$\displaystyle 6$

Correct answer:

$\displaystyle 10$

Explanation:

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

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

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

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Example Question #392 : Geometry

What is the distance between (1,-3) $\displaystyle (1,-3)$ and (13,-8) $\displaystyle (13,-8)$ ?

Possible Answers:

$\displaystyle 16$

$\displaystyle 12$

$\displaystyle 13$

$\displaystyle 15$

$\displaystyle 14$

Correct answer:

$\displaystyle 13$

Explanation:

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

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

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

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Example Question #2 : Lines

What is the distance between the points (-2,1) $\displaystyle (-2,1)$ and (10,-4) $\displaystyle (10,-4)$?

Possible Answers:

$\displaystyle 13$

$\displaystyle 12$

$\displaystyle 11$

$\displaystyle 14$

$\displaystyle 10$

Correct answer:

$\displaystyle 13$

Explanation:

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

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

Plug these two points into the distance formula:

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

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Example Question #1 : How To Find Length Of A Line

If a line with a length of 5 starts at the point (1,1) $\displaystyle (1,1)$, which of the following is NOT a possible end point?

Possible Answers:

(-3,5) $\displaystyle (-3,5)$

(5, -2) $\displaystyle (5, -2)$

$\displaystyle (-2, -3)$

(-3, 4) $\displaystyle (-3, 4)$

(4, 5) $\displaystyle (4, 5)$

Correct answer:

(-3,5) $\displaystyle (-3,5)$

Explanation:

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

Recall the distance formula:

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

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

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

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

Substitute each response into this equation to see which one is false. The only response that doesn't work is (-3, 5) $\displaystyle (-3, 5)$.

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Example Question #1 : How To Find Length Of A Line

What is the distance between (-2, 1) $\displaystyle (-2, 1)$ and (1,5) $\displaystyle (1,5)$?

Possible Answers:

$\displaystyle 9$

$\displaystyle 7$

$\displaystyle 5$

$\displaystyle 8$

$\displaystyle 6$

Correct answer:

$\displaystyle 5$

Explanation:

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

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

Plug these points into the distance formula:

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

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Example Question #401 : Geometry

If a line that is 13 inches in length starts at the point (1,2) $\displaystyle (1,2)$, which of the following is NOT a possible end point?

Possible Answers:

(6, 14) $\displaystyle (6, 14)$

(13, 7) $\displaystyle (13, 7)$

(3, 15) $\displaystyle (3, 15)$

$\displaystyle (-4, -10)$

$\displaystyle (-11, -3)$

Correct answer:

(3, 15) $\displaystyle (3, 15)$

Explanation:

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

Square both sides:

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

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

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

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

Alternatively, you can plug in the answer choices and see which point makes the above equation untrue. The point (3, 15) $\displaystyle (3, 15)$ matches this description and is therefore the correct answer.

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ISEE Lower Level Math : Lines

Study concepts, example questions & explanations for ISEE Lower Level Math

All ISEE Lower Level Math Resources

Example Questions

Example Question #1 : Lines

Example Question #1 : Lines

Example Question #1 : Lines

Example Question #391 : Geometry

Example Question #1331 : Isee Lower Level (Grades 5 6) Mathematics Achievement

Example Question #392 : Geometry

Example Question #2 : Lines

Example Question #1 : How To Find Length Of A Line

Example Question #1 : How To Find Length Of A Line

Example Question #401 : Geometry

All ISEE Lower Level Math Resources

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