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

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

$2\sqrt3$

$6\sqrt2$

$3\sqrt3$

$3\sqrt2$

$\frac{3\sqrt2}{2}$

Explanation

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, .

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

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

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

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

Undetermined

True

False

Explanation

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.

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

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

False

True

Undetermined

Explanation

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.

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.

$\sqrt{142}cm$

$2\sqrt{38}cm$

None of the other answers.

$4\sqrt{38}cm$

$3\sqrt{3}cm$

Explanation

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.

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

None of the other answers.

Explanation

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 .