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

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Example Question #193 : Sat Mathematics

What is the length of the diagonal of a rectangle that is 3 feet long and 4 feet wide?

Possible Answers:

$8\ feet$ $\displaystyle 8\ feet$

$6\ feet$ $\displaystyle 6\ feet$

$5\ feet$ $\displaystyle 5\ feet$

$4\ feet$ $\displaystyle 4\ feet$

$7\ feet$ $\displaystyle 7\ feet$

Correct answer:

$5\ feet$ $\displaystyle 5\ feet$

Explanation:

The diagonal of the rectangle is equivalent to finding the length of the hypotenuse of a right triangle with sides 3 and 4. Using the Pythagorean Theorem:

$3^{2}+4^{2} = hypotenuse^{2}$ $\displaystyle 3^{2}+4^{2} = hypotenuse^{2}$

$25 = hypotenuse^{2}$ $\displaystyle 25 = hypotenuse^{2}$

$hypotenuse = 5$ $\displaystyle hypotenuse = 5$

Therefore the diagonal of the rectangle is 5 feet.

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Example Question #2 : How To Find The Length Of The Diagonal Of A Rectangle

The length and width of a rectangle are in the ratio of 3:4. If the rectangle has an area of 108 square centimeters, what is the length of the diagonal?

Possible Answers:

24 centimeters

9 centimeters

15 centimeters

18 centimeters

12 centimeters

Correct answer:

15 centimeters

Explanation:

The length and width of the rectangle are in a ratio of 3:4, so the sides can be written as 3x and 4x.

We also know the area, so we write an equation and solve for x:

(3x)(4x) = 12x²= 108.

x² = 9

x = 3

Now we can recalculate the length and the width:

length = 3x = 3(3) = 9 centimeters

width = 4x = 4(3) = 12 centimeters

Using the Pythagorean Theorem we can find the diagonal, c:

length² + width² = c²

9² + 12²= c²

81 + 144 = c²

225 = c²

c = 15 centimeters

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Example Question #2 : How To Find The Length Of The Diagonal Of A Rectangle

Find the length of the diagonal of a rectangle whose sides are 8 and 15.

Possible Answers:

$\sqrt{17}$ $\displaystyle \sqrt{17}$

$\displaystyle 17$

$\displaystyle 23$

$\sqrt{23}$ $\displaystyle \sqrt{23}$

Correct answer:

$\displaystyle 17$

Explanation:

To solve. simply use the Pythagorean Theorem where a=8 $\displaystyle a=8$ and b=15 $\displaystyle b=15$.

Thus,

$c=\sqrt{a^2+b^2}=\sqrt{8^2+15^2}=\sqrt{289}=17$ $\displaystyle c=\sqrt{a^2+b^2}=\sqrt{8^2+15^2}=\sqrt{289}=17$

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Example Question #202 : Geometry

Prism

The above figure depicts a cube, each edge of which has length 18. Give the length of the shortest path from Point A to Point B that lies completely along the surface of the cube.

Possible Answers:

$18\sqrt{5}$ $\displaystyle 18\sqrt{5}$

$18 \sqrt{2}$ $\displaystyle 18 \sqrt{2}$

$18\sqrt{3}$ $\displaystyle 18\sqrt{3}$

$\displaystyle 36$

$18\sqrt{6}$ $\displaystyle 18\sqrt{6}$

Correct answer:

$18\sqrt{5}$ $\displaystyle 18\sqrt{5}$

Explanation:

The shortest path is along two of the surfaces of the prism. There are three possible choices - top and front, right and front, and rear and bottom - but as it turns out, since all faces are (congruent) squares, all three paths have the same length. One such path is shown below, with the relevant faces folded out:

Prism 2

The length of the path can be seen to be equal to that of the diagonal of a rectangle with length and width 18 and 36, so its length can be found by applying the Pythagorean Theorem. Substituting 18 and 36 for $\displaystyle a$ and $\displaystyle b$:

$c = \sqrt{a^{2}+b^{2}}$ $\displaystyle c = \sqrt{a^{2}+b^{2}}$

$= \sqrt{18^{2}+36^{2}}$ $\displaystyle = \sqrt{18^{2}+36^{2}}$

$= \sqrt{324+1,296}$ $\displaystyle = \sqrt{324+1,296}$

$= \sqrt{1,620 }$ $\displaystyle = \sqrt{1,620 }$

Applying the Product of Radicals Rule:

$c = \sqrt{1,620 } = \sqrt{324} \cdot \sqrt{5 } = 18 \sqrt{5 }$ $\displaystyle c = \sqrt{1,620 } = \sqrt{324} \cdot \sqrt{5 } = 18 \sqrt{5 }$.

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

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #193 : Sat Mathematics

Example Question #2 : How To Find The Length Of The Diagonal Of A Rectangle

Example Question #2 : How To Find The Length Of The Diagonal Of A Rectangle

Example Question #202 : Geometry

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