Perimeter - SAT Math

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Question

Note: Figure NOT drawn to scale.

Refer to the figure above, which shows a square inscribed inside a large triangle. What is the difference between the perimeter of the white trapezoid and the blue triangle?

Answer

One side of the white trapezoid is the hypotenuse of the small top triangle, which has legs 10 and 20. Therefore, the length of this side can be determined using the Pythagorean Theorem:

$c = \sqrt{10^{2}+ 20^{2}} = \sqrt{100 + 400} = \sqrt{500} = \sqrt{100} \cdot \sqrt{5} = 10 \sqrt{5}$

The trapezoid has perimeter

$30 + 20 + 20 + 10 \sqrt{5} = 70 + 10 \sqrt{5}$ .

The small top triangle has legs 10 and 20 and hypotenuse $10 \sqrt{5}$ , and, therefore, perimeter

$10 + 20 + 10 \sqrt{5} = 30 + 10 \sqrt{5}$ . The blue triangle, which is similar as a result of the parallelism of the opposite sides of the square, has short leg 20. Since the perimeter of two similar triangles is directly proportional to a side, we can set up and solve the proportion statement to find the perimeter of the blue triangle:

$\frac{P}{30 + 10 \sqrt{5}} = \frac{20}{10}$

$\frac{P}{30 + 10 \sqrt{5}}\left ( 30 + 10 \sqrt{5} \right ) = \frac{20}{10}\left ( 30 + 10 \sqrt{5} \right )$

$P =60 + 20 \sqrt{5}$

The difference between the perimeters is

$\left (60 + 20 \sqrt{5} \right )- \left (70 + 10 \sqrt{5} \right )=-10 +10 \sqrt{5}$

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Question

Refer to the above figure. Quadrilateral is a square. Give the perimeter of Polygon in terms of .

Answer

$\overline{BE }$ is both one side of Square and the hypotenuse of $\Delta ABE$ ; its hypotenuse can be calculated from the lengths of the legs using the Pythagorean Theorem:

$BE = \sqrt{(AB)^{2}+ \left ( AE\right )^{2}}$

$BE = \sqrt{x^{2}+ \left ( 2x\right )^{2}} = \sqrt{x^{2}+ 4x ^{2}} = \sqrt{5x ^{2}} = x\sqrt{5}$

Therefore, $BC = CD = DE = x\sqrt{5}$ .

The perimeter of Polygon is

$P=x + x\sqrt{5} + x\sqrt{5} + x\sqrt{5}+ 2x$

$P=3x + 3x\sqrt{5} = \left ( 3 + 3\sqrt{5}\right )x$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the ratio of the perimeter of $\Delta BDC$ to that of $\Delta ADB$ .

Answer

The altitude of a right triangle from the vertex of its right triangle to its hypotenuse divides it into two similar triangles.

, as the length of the altitude corresponding to the hypotenuse, is the geometric mean of the lengths of the parts of the hypotenuse it forms; that is, it is the square root of the product of the two:

$BD =\sqrt{ \left ( AD\right ) \left ( DC\right )} = \sqrt{6 \cdot 24} = \sqrt{144} = 12$ .

The ratio of the smaller side of $\Delta BDC$ to that of $\Delta ADB$ is

$\frac{BD}{AD}=\frac{12}{6} = \frac{2}{1}$ or 2:1, making this the similarity ratio. Theratio of the perimeters is always equal to the similarity ratio, so 2:1 is also the ratio of the perimeter of $\Delta BDC$ to that of $\Delta ADB$ .

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Question

The area of a rectangle is 16. Assuming the length and width are integers, which of the answers is NOT a possible perimeter?

Answer

The area of a rectangle is length times width.

Determine all the integer combinations that will multiply to an area of 16. The numbers can represent length or width interchangably.

Write the perimeter formula for a rectangle.

Substitute all the following combinations to determine the perimeters.

The maximum allowable perimeter is

The perimeter that is not possible is .

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Question

If a side of a square has a length of , what is the perimeter?

Answer

Step 1: Define Perimeter...

The perimeter of any shape is the sum of all the sides of that figure..

Step 2: Find how many sides a square has...

A square has sides.

Step 3: Calculate the perimeter...

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Question

What is the perimeter of a rectangle if the area is 60, and one of the side length is 15?

Answer

Write the formula for the area of the rectangle. We will need the length of the other side of the rectangle.

Let the known side length be the base.

Divide by 15 on both sides to get the height.

$\frac{60}{15}=\frac{15h}{15}$

There are two bases and two heights in a rectangle.

The perimeter is:

The answer is:

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Question

Find the perimeter of an equilateral triangle with a length of .

Answer

An equilateral triangle has three congruent side lengths.

This means that the perimeter has to be triple the side length.

Distribute the three with both terms of the binomial.

The answer is:

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Question

Refer to the above figure. Quadrilateral is a square. What is the perimeter of Polygon ?

Answer

$\overline{BE }$ is both one side of Square and the hypotenuse of $\Delta ABE$ ; its hypotenuse can be calculated from the lengths of the legs using the Pythagorean Theorem:

$BE = \sqrt{5^{2}+12^{2}} = \sqrt{25 + 144} = \sqrt{169} = 13$ .

Since Square has four congruent sides, each side has length 13.

The perimeter of Polygon is

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Question

Note: Figure NOT drawn to scale

Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in orange) seven feet wide throughout. What is the perimeter of the garden?

Answer

The inner rectangle, which represents the garden, has length and width $(100 - 2\times 7) = 86$ feet and $(60 - 2\times 7) = 46$ feet, respectively, so its perimeter is

feet.

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Question

Refer to the above figure, which shows a square garden (in green) surrounded by a dirt path (in orange). The dirt path is seven feet wide throughout. Which of the following polynomials gives the perimeter of the garden in feet?

Answer

The sidelength of the garden, in feet, is $2 \times 7 = 14$ feet less than that of the entire lot, or

;

The perimeter, in feet, of the garden is four times this:

$P = 4S = 4 \left (x-14 \right ) = 4x - 56$

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Question

You have a pentagonal-shaped lot with side lengths of 120ft ,30ft, 55ft, 60ft, and a longest side which is triple the length of the third longest side. What is the perimeter around the lot?

Answer

You have a pentagonal-shaped lot with side lengths of 120ft,30ft, 55ft, 60 ft, and a longest side which is triple the length of the third longest side. What is the perimeter around the lot?

So, we have a five-sided lot and are given 4 sides, and the means to find the 5th.

The longest side is triple the length of the third longest side:

The third longest side is 60 feet, therefore, the longest side is 180 ft

Find the perimeter by adding up all the sides:

So our answer is 445 feet

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Question

The image represents a track is a regular hexagon with perimeter one half of a mile.

Selena starts at Point A and runs clockwise to Point D. She then runs directly from Point D to Point A. How far does she run?

Answer

The perimeter of the hexagonal track is one half of a mile. One mile is equal to 5,280 feet, so one half of a mile is equal to

$\frac{1}{2} \times 5,280 = 2,640\textup{ feet}$

Each of the six congruent sides measures one sixth of this, or

$\frac{1}{6} \times 2,640 = 440\textup{ feet}$

The hexagonal track is recreated below, with its six radii constructed - the center is called .

The radii divide the hexagon into six equilateral triangles, so each of the radii has length equal to one side of the hexagon. Selena's path takes her along $\overline{AB}$ , $\overline{BC}$ , $\overline{CD}$ , $\overline{DO}$ , and $\overline{OA}$ - a path equivalent in distance to five sides of the hexagon. Therefore, Selena runs a distance of

$5 \times 440 = 2,200 \textup{ feet}$

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Question

Find the circumference of a circle with diameter of 15.

Answer

The circumference of a circle is denoted by the formula $C=2\pir$ where r is the radius of the circle. The problem gives us the diameter of the circle which must be used to find the radius.

$radius=\frac{Diameter}{2}=\frac{15}{2}=7.5$

Plug the radius into the formula for the area of a circle

$C=2\pi7.5=15\pi$

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Question

If the sides of a triangle are , , and , what is the perimeter?

Answer

The perimeter of a triangle is the sum of all three sides of the triangle.

Add all the side lengths.

Expand the terms in the bracket and remove all parentheses.

Combine like-terms.

The answer is:

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Question

If the side length of a rectangle is 6 inches, what is the perimeter?

Answer

A square can be a rectangle, but a rectangle cannot be a square. We cannot assume that the width is known given just the length of the rectangle. There is not enough information to determine the perimeter just by the side length alone.

The answer is: $\textup{Cannot be determined.}$

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Question

What is the perimeter of a square if the area is units squared?

Answer

Determine the length of the square by using the given area.

Substitute the area.

$s=\sqrt{10}$

There are four congruent sides in a square, which means the perimeter is four times the side.

The answer is: $4\sqrt{10}$

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Question

What is the perimeter of a right triangle with a base of 6, and a height of 8?

Answer

Use the Pythagorean Theorem to determine the hypotenuse.

Substitute the base and height.

Square root both sides to find the hypotenuse.

Add the sides to determine the perimeter.

The answer is:

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Question

Note: Figure NOT drawn to scale.

Refer to the figure above, which shows a square inscribed inside a large triangle. What is the difference between the perimeter of the white trapezoid and the blue triangle?

Answer

One side of the white trapezoid is the hypotenuse of the small top triangle, which has legs 10 and 20. Therefore, the length of this side can be determined using the Pythagorean Theorem:

$c = \sqrt{10^{2}+ 20^{2}} = \sqrt{100 + 400} = \sqrt{500} = \sqrt{100} \cdot \sqrt{5} = 10 \sqrt{5}$

The trapezoid has perimeter

$30 + 20 + 20 + 10 \sqrt{5} = 70 + 10 \sqrt{5}$ .

The small top triangle has legs 10 and 20 and hypotenuse $10 \sqrt{5}$ , and, therefore, perimeter

$10 + 20 + 10 \sqrt{5} = 30 + 10 \sqrt{5}$ . The blue triangle, which is similar as a result of the parallelism of the opposite sides of the square, has short leg 20. Since the perimeter of two similar triangles is directly proportional to a side, we can set up and solve the proportion statement to find the perimeter of the blue triangle:

$\frac{P}{30 + 10 \sqrt{5}} = \frac{20}{10}$

$\frac{P}{30 + 10 \sqrt{5}}\left ( 30 + 10 \sqrt{5} \right ) = \frac{20}{10}\left ( 30 + 10 \sqrt{5} \right )$

$P =60 + 20 \sqrt{5}$

The difference between the perimeters is

$\left (60 + 20 \sqrt{5} \right )- \left (70 + 10 \sqrt{5} \right )=-10 +10 \sqrt{5}$

Compare your answer with the correct one above

Question

Refer to the above figure. Quadrilateral is a square. Give the perimeter of Polygon in terms of .

Answer

$\overline{BE }$ is both one side of Square and the hypotenuse of $\Delta ABE$ ; its hypotenuse can be calculated from the lengths of the legs using the Pythagorean Theorem:

$BE = \sqrt{(AB)^{2}+ \left ( AE\right )^{2}}$

$BE = \sqrt{x^{2}+ \left ( 2x\right )^{2}} = \sqrt{x^{2}+ 4x ^{2}} = \sqrt{5x ^{2}} = x\sqrt{5}$

Therefore, $BC = CD = DE = x\sqrt{5}$ .

The perimeter of Polygon is

$P=x + x\sqrt{5} + x\sqrt{5} + x\sqrt{5}+ 2x$

$P=3x + 3x\sqrt{5} = \left ( 3 + 3\sqrt{5}\right )x$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the ratio of the perimeter of $\Delta BDC$ to that of $\Delta ADB$ .

Answer

The altitude of a right triangle from the vertex of its right triangle to its hypotenuse divides it into two similar triangles.

, as the length of the altitude corresponding to the hypotenuse, is the geometric mean of the lengths of the parts of the hypotenuse it forms; that is, it is the square root of the product of the two:

$BD =\sqrt{ \left ( AD\right ) \left ( DC\right )} = \sqrt{6 \cdot 24} = \sqrt{144} = 12$ .

The ratio of the smaller side of $\Delta BDC$ to that of $\Delta ADB$ is

$\frac{BD}{AD}=\frac{12}{6} = \frac{2}{1}$ or 2:1, making this the similarity ratio. Theratio of the perimeters is always equal to the similarity ratio, so 2:1 is also the ratio of the perimeter of $\Delta BDC$ to that of $\Delta ADB$ .

Compare your answer with the correct one above

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