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SAT Math : How to find the length of the side of a rectangle

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Example Question #1 : Rectangles

The two rectangles shown below are similar. What is the length of EF?

Sat_mah_166_02

Possible Answers:

Correct answer:

Explanation:

When two polygons are similar, the lengths of their corresponding sides are proportional to each other. In this diagram, AC and EG are corresponding sides and AB and EF are corresponding sides.

To solve this question, you can therefore write a proportion:

AC/EG = AB/EF ≥ 3/6 = 5/EF

From this proportion, we know that side EF is equal to 10.

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Example Question #1 : How To Find The Length Of The Side Of A Rectangle

A rectangle is x inches long and 3x inches wide. If the area of the rectangle is 108, what is the value of x?

Possible Answers:

Correct answer:

Explanation:

Solve for x

Area of a rectangle A = lw = x(3x) = 3x² = 108

x² = 36

x = 6

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Example Question #1 : How To Find The Length Of The Side Of A Rectangle

If the area of rectangle is 52 meters squared and the perimeter of the same rectangle is 34 meters. What is the length of the larger side of the rectangle if the sides are integers?

Possible Answers:

14 meters

15 meters

16 meters

12 meters

13 meters

Correct answer:

13 meters

Explanation:

Area of a rectangle is = lw

Perimeter = 2(l+w)

We are given 34 = 2(l+w) or 17 = (l+w)

possible combinations of l + w

are 1+16, 2+15, 3+14, 4+13... ect

We are also given the area of the rectangle is 52 meters squared.

Do any of the above combinations when multiplied together= 52 meters squared? yes 4x13 = 52

Therefore the longest side of the rectangle is 13 meters

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Example Question #33 : Quadrilaterals

Rectangles a

Figure is not drawn to scale.

The provided figure is a rectangle divided into two smaller rectangles, with

Rectangle $ABEF \sim$ Rectangle CDEB .

Which expression is equal to the length of $\overline{FE}$ ?

Possible Answers:

$12+ 2 \sqrt{5}$

$12 + 4\sqrt{5}$

$12+\sqrt{5}$

Correct answer:

$12 + 4\sqrt{5}$

Explanation:

Since Rectangle ABEF is similar to Rectangle CDEB , it follows that corresponding sides are in proportion. Specifically,

$\frac{FE}{BE} = \frac{BE}{DE}$

FE + DE = FD ;

since FD = 24 , if we let t = FE , then

t+ DE = 24 ,

and

DE= 24 - t

Setting BE = 8 , FE = t , and DE= 24 - t , the proportion statement becomes

$\frac{t}{8} = \frac{8}{24-t }$

Cross-multiplying, we get

$t(24-t) = 8 \cdot 8$

Simplifying, we get

$24t-t ^{2}= 64$

Since this is quadratic, all terms must be moved to one side:

$24t-t ^{2} - 24t + t ^{2} = 64 - 24t + t ^{2}$

$0= 64 - 24t + t ^{2}$

$t ^{2} - 24t + 64 = 0$

The solutions to quadratic equation

$ax^{2} + bx + c = 0$

can be found by way of the quadratic formula

$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

Set a = 1, b= -24, c= 64 :

$x = \frac{-(-24) \pm \sqrt{(-24)^{2}-4(1)(64)}}{2(1)}$

$= \frac{ 24 \pm \sqrt{576-256}}{2}$

$= \frac{ 24 \pm \sqrt{320}}{2}$

Simplifying the radical using the Product of Radicals Property, we get

$\frac{ 24 \pm \sqrt{64}\cdot \sqrt{5}}{2}$

$=\frac{ 24 \pm8\sqrt{5}}{2}$

Splitting the fraction and reducing:

$\frac{ 24 }{2} \pm \frac{ 8\sqrt{5}}{2}$

$=12 \pm 4\sqrt{5}$

This actually tells us that the lengths of the two segments $\overline{FE}$ and $\overline{ED}$ have the lengths $12 + 4\sqrt{5}$ and $12 -4\sqrt{5}$ . $\overline{FE}$ is seen in the diagram to be the longer, so we choose the greater value, $12 + 4\sqrt{5}$ .

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SAT Math : How to find the length of the side of a rectangle

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : Rectangles

Example Question #1 : How To Find The Length Of The Side Of A Rectangle

Example Question #1 : How To Find The Length Of The Side Of A Rectangle

Example Question #33 : Quadrilaterals

All SAT Math Resources

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