How to find the length of the diagonal of a rectangle - Basic Geometry

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Question

A rectangle has a length of 12 in and a width of 5 in. What is the diagonal of the rectangle?

Answer

To find the diagonal we use the Pythagorean Theorem:

$5^{2}+12^{2}= x^{2}$ where = hypotenuse

$25+144=x^{2}$ or

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Question

Find the length of the diagonal.

Answer

The rectangle can be cut into two equal right triangles, where the hypotenuse of both is the rectangle's diagonal.

Use the Pythagorean Theorem to find the hypotenuse:

, where a and b are the legs and c is the hypotenuse

$\sqrt{208} = \sqrt{c^2}$

$c = 4\sqrt{13}$

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Question

One side of a rectangle is 7 inches and another is 9 inches. How many inches long is the rectangle's diagonal?

Answer

You can find the diagonal of a rectangle if you have the width and the height. The diagonal equals the square root of the width squared plus the height squared.

$\sqrt{9^{2}+7^{2}}=11.4$

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Question

A rectangle has a height of and a base of . What is the length of its diagonal rounded to the nearest tenth?

Answer

1. Use Pythagorean Theorem with and .

$a^{2}+b^{2}=c^{2}$

$14^{2}+20^{2}=c^{2}$

2. Solve for , the length of the diagonal:

$14^{2}+20^{2}=c^{2}$

$196+400=c^{2}$

$596=c^{2}$

This rounds down to because the hundredth's place () is less than .

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Question

The sides of rectangle ABCD are 4 in and 13 in.

How long is the diagonal of rectangle ABCD?

Answer

A diagonal of a rectangle cuts the rectangle into 2 right triangles with sides equal to the sides of the rectangle and with a hypotenuse that is the diagonal. All you need to do is use the pythagorean theorem:

where a and b are the sides of the rectangle and c is the length of the diagonal.

$\sqrt{4^2+16^2}=\sqrt{185}=c$

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Question

A standard high school basketball court is 84 feet long and 50 feet wide. During practice, Coach K has Kyrie run from one the right corner on one end of the court to the left corner at the other end of the court. To the nearest foot, how far did Kyrie run?

Answer

A picture helps greatly with this problem, so we begin with a rectangular basketball court.

We note that the distance run by Kyrie (drawn in red) is the diagonal of our rectangle, which we will call . We should also not that this diagonal divides our rectangle into two congruent right triangles. We can therefore find the length of our diagonal by focusing on one of these triangles and determining the hypotenuse. This can be done with the Pythagorean Theorem, which gives us:

Taking the square root gives us

Rounding to the nearest foot gives an answer of 98.

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Question

Find the length of the diagonal of a rectangle that has a length of and a width of .

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{9^2+12^2}=\sqrt{225}=15$

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Question

Find the length of the diagonal of a rectangle that has a length of and a width of .

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{10^2+24^2}=\sqrt{676}=26$

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Question

Find the length of the diagonal of a rectangle that has a length of and a width of .

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{6^2+8^2}=\sqrt{100}=10$

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Question

Find the length of the diagonal of a rectangle that has a length of and a width of .

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{9^2+15^2}=\sqrt{306}=3\sqrt{34}$

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Question

Find the length of the diagonal of a rectangle that has a length of and a width of .

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{13^2+84^2}=\sqrt{7225}=85$

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Question

Find the length of the diagonal of a rectangle that has a length of and a width of .

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{2^2+3^2}=\sqrt{13}$

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Question

Find the length of the diagonal of a rectangle that has a length of and a width of .

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{8^2+2^2}=\sqrt{68}=2\sqrt{17}$

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Question

Find the length of the diagonal of a rectangle that has a length of and a width of .

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{5^2+2^2}=\sqrt{29}$

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Question

Find the length of the diagonal of a rectangle that has a length of and a width of .

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{4^2+10^2}=\sqrt{116}=2\sqrt{29}$

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Question

Find the length of the diagonal of a rectangle that has a length of and a width of .

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{10^2+11^2}=\sqrt{221}$

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Question

Find the length of the diagonal of a rectangle that has a width of and length of ?

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{14^2+12^2}=\sqrt{340}=2\sqrt{85}$

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Question

Find the length of the diagonal of a rectangle that has a length of and a width of .

Answer

The diagonal of a rectangle is also the hypotenuse of a right triangle that has the length and the width of the rectangle as its legs.

We can then use the Pythgorean Theorem to find the diagonal.

$\text{Diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

For the given rectangle,

$\text{Diagonal}=\sqrt{4^2+16^2}=\sqrt{272}=4\sqrt{17}$

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Question

Find the length of the diagonal of a rectangle whose length is 5 and width is 3.

Answer

To solve, simply use the pythagorean theorem. Thus,

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

$c=\sqrt{5^2+3^2}=\sqrt{25+9}=\sqrt{34}$

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Question

A rectangle is inscribed in a circle. If the circumference of the circle is $2\pi$ , what is the length of the diagonal of the rectangle?

Answer

You should notice that the diagonal of the rectangle is the same as the diameter of the circle.

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\text{diameter}\times\pi$

Since we are given the circumference and need the diameter, rerwite the equation to solve for the diameter.

$\text{Diameter}=\frac{\text{Circumference}}{\pi}$

Plug in the given circumference to find the diameter.

$\text{Diameter}=\frac{2\pi}{\pi}=2$

Now, recall the relationship between the diameter of the circle and the diagonal of the rectangle:

$\text{Diameter}=\text{Diagonal}=2$

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