How to find the length of the diagonal of a parallelogram - Intermediate Geometry

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Question

The length of a parallelogram is cm and the width is cm. One of its diagonals measures cm. Find the length of the other diagonal.

Answer

The formula for the relationship between diagonals and sides of a parallologram is

$(d_{1})^{2} + (d_{2})^{2} = 2a^{2} + 2b^{2}$ ,

where $d_{1}$ represents one diagonal,

$d_{2}$ represents the other diagonal,

represents a side, and

represents the adjoining side.

So, in this problem, substitute the known values and solve for the missing diagonal.

$14^{2} + (d_{2})^{2} = 2(12)^{2} + 2(5)^{2} \rightarrow 196 + d^{2} = 288 + 50$

$\rightarrow d^{2} = 142 \rightarrow d = \sqrt{142}$

So, the missing diagonal is $\sqrt{142}$ cm.

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Question

In the parallogram above, find the length of the labeled diagonal.

Answer

In a parallogram, diagonals bisect one another, thus you can set the two segments that are labeled in the picture equal to one another, then solve for .

So,

$2x - 2 = x + 12 \rightarrow x = 14$ .

If , then you can substitute 14 into each labeled segment, to get a total of 52.

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Question

In the parallogram below, find the length of the labeled diagonal.

Answer

In a parallelogram, the diagonals bisect one another, so you can set the labeled segments equal to each other and then solve for .

$x + 14 = 5x - 10 \rightarrow 24 = 4x \rightarrow 6 = x$ .

If , then you substitute 6 into each labeled segment, to get a total of 40.

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Question

In the parallelogram above, find the length of the labeled diagonal.

Answer

In a parallelogram, the diagonals bisect each other, so you can set the labeled segments equal to one another and then solve for .

$20 - 3x = 2x - 4 \rightarrow 24 = 5x \rightarrow 4.8 = x$ .

Then, substitute 4.8 for in each labeled segment to get a total of 11.2 for the diagonal length.

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Question

Suppose a square has an area of 6. What is the diagonal of the parallelogram?

Answer

Write the formula to find the side of the square given the area.

Find the side.

$s=\sqrt6$

The diagonal of the square can be solved by using the Pythagorean Theorem.

$a=b=s=\sqrt6$

Substitute and solve for the diagonal, .

$(\sqrt6)^2+(\sqrt6)^2 =c^2$

$c=\sqrt{12}=2\sqrt3$

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Question

If the side length of a square is , what is the diagonal of the square?

Answer

Write the diagonal formula for a square.

$d=s\sqrt2$

Substitute the side length and reduce.

$d=3a\sqrt2$

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Question

Parallelogram has diagonals $\overline{AC}$ and $\overline{BD}$ . and .

True, false, or undetermined: Parallelogram is a rectangle.

Answer

One characteristic of a rectangle is that its diagonals are congruent. Since the diagonals of Parallelogram are of different lengths, it cannot be a rectangle.

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Question

Parallelogram has diagonals $\overline{AC}$ and $\overline{BD}$ . and .

True, false, or undetermined: Parallelogram is a rhombus.

Answer

One characteristic of a rhombus is that its diagonals are perpendicular; no restrictions exist as to their lengths. Whether or not the diagonals are perpendicular is not stated, so the figure may or may not be a rhombus.

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