Plane Geometry - ISEE Lower Level (grades 5-6) Mathematics Achievement

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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

How many lines of symmetry are there in a square?

Answer

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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Question

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?

Answer

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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Question

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

Answer

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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Question

Find the area of the parallelogram:

Answer

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Question

What is the area of the parallelogram ABCD?

Answer

A parallelogram's area is found by multiplying its height by the base. The height of the parallelogram is not the side. It is the line that makes a right angle with the base. Therefore, the area of this parallelogram is:

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Question

What is the area of parallelogram ABCD?

Answer

A parallelogram's area is found by multiplying its height by the base. The height of the parallelogram is not the side. It is the line that makes a right angle with the base. Therefore, the area of this parallelogram is:

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Question

A parallelogram measures along its base and is high. Calculate the area.

Answer

The formula to calucate a parallelogram's area is:

$A = B \times H$

You calculate the parallelogram's area by multiplying its base by its height. Therefore, $A = 4 cm \times 7 cm$

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Question

A parallelogram measures along its base and is high. Calculate the area.

Answer

The formula to calucate a parallelogram's area is:

$A = B\times H$

You calculate the parallelogram's area by multiplying its base by its height. Therefore, $A = 6 cm \times 9 cm$

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Question

Find the area of a parallelogram whose height is and base is .

Answer

To solve, simply use the formula for the area of a parallelogram. Thus,

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Question

What is the area of a parallelogram if the base is , the other side is , and the height is ?

Answer

The area of a parallelogram is so the answer would be .

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Question

Find the area of the given parallelogram:

Answer

Find the area of the given parallelogram:

To find the area of a parallelogram, simply do the following:

Where b is the base, and h is the height. Note that the height is the length of the perpendicular line connecting both bases. In this case, our height is 12mi and our base is 144 miles.

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