How to find the length of the diagonal of a rectangle - SAT Math
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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?
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?
The length and width of the rectangle are in a ratio of 3:4, so the sides can be written as 3_x_ and 4_x_.
We also know the area, so we write an equation and solve for x:
(3_x_)(4_x_) = 12_x_2 = 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 = _c_2
81 + 144 = c2
225 = c2
c = 15 centimeters
The length and width of the rectangle are in a ratio of 3:4, so the sides can be written as 3_x_ and 4_x_.
We also know the area, so we write an equation and solve for x:
(3_x_)(4_x_) = 12_x_2 = 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 = _c_2
81 + 144 = c2
225 = c2
c = 15 centimeters
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What is the length of the diagonal of a rectangle that is 3 feet long and 4 feet wide?
What is the length of the diagonal of a rectangle that is 3 feet long and 4 feet wide?
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:
Therefore the diagonal of the rectangle is 5 feet.
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:
Therefore the diagonal of the rectangle is 5 feet.
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Find the length of the diagonal of a rectangle whose sides are 8 and 15.
Find the length of the diagonal of a rectangle whose sides are 8 and 15.
To solve. simply use the Pythagorean Theorem where 
and 
.
Thus,
To solve. simply use the Pythagorean Theorem where
Thus,
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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.
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.
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:
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 
and 
:
Applying the Product of Radicals Rule:
.
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:
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
Applying the Product of Radicals Rule:
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