Intermediate Geometry : How to find the length of the side of a rhombus

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #31 : How To Find The Length Of The Side Of A Rhombus

Find the length of a side of a rhombus that has diagonals with lengths of \(\displaystyle 40\) and \(\displaystyle 46\).

Possible Answers:

\(\displaystyle 33.04\)

\(\displaystyle 32.19\)

\(\displaystyle 30.48\)

\(\displaystyle 28.45\)

Correct answer:

\(\displaystyle 30.48\)

Explanation:

13

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

\(\displaystyle \text{Half Diagonal 1}=\frac{40}{2}=20\)

\(\displaystyle \text{Half Diagonal 2}=\frac{46}{2}=23\)

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

\(\displaystyle \text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}\)

Plug in the lengths of the half diagonals to find the length of the rhombus.

\(\displaystyle \text{Side of Rhombus}=\sqrt{20^2+23^2}=\sqrt{929}=30.48\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

Example Question #32 : How To Find The Length Of The Side Of A Rhombus

Find the length of a side of a rhombus that has diagonals with lengths of \(\displaystyle 42\) and \(\displaystyle 50\).

Possible Answers:

\(\displaystyle 38.02\)

\(\displaystyle 32.65\)

\(\displaystyle 30.01\)

\(\displaystyle 29.95\)

Correct answer:

\(\displaystyle 32.65\)

Explanation:

13

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

\(\displaystyle \text{Half Diagonal 1}=\frac{42}{2}=21\)

\(\displaystyle \text{Half Diagonal 2}=\frac{50}{2}=25\)

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

\(\displaystyle \text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}\)

Plug in the lengths of the half diagonals to find the length of the rhombus.

\(\displaystyle \text{Side of Rhombus}=\sqrt{21^2+25^2}=\sqrt{1066}=32.65\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

Example Question #33 : How To Find The Length Of The Side Of A Rhombus

Find the length of a side of a rhombus that has diagonals with lengths of \(\displaystyle 90\) and \(\displaystyle 100\).

Possible Answers:

\(\displaystyle 68.09\)

\(\displaystyle 72.04\)

\(\displaystyle 67.27\)

\(\displaystyle 69.55\)

Correct answer:

\(\displaystyle 67.27\)

Explanation:

13

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

\(\displaystyle \text{Half Diagonal 1}=\frac{90}{2}=45\)

\(\displaystyle \text{Half Diagonal 2}=\frac{100}{2}=50\)

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

\(\displaystyle \text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}\)

Plug in the lengths of the half diagonals to find the length of the rhombus.

\(\displaystyle \text{Side of Rhombus}=\sqrt{45^2+50^2}=\sqrt{4525}=67.27\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

Example Question #34 : How To Find The Length Of The Side Of A Rhombus

Find the length of a side of a rhombus that has diagonals with lengths of \(\displaystyle 52\) and \(\displaystyle 58\).

Possible Answers:

\(\displaystyle 37.55\)

\(\displaystyle 39.05\)

\(\displaystyle 38.95\)

\(\displaystyle 40.12\)

Correct answer:

\(\displaystyle 38.95\)

Explanation:

13

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

\(\displaystyle \text{Half Diagonal 1}=\frac{52}{2}=26\)

\(\displaystyle \text{Half Diagonal 2}=\frac{58}{2}=29\)

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

\(\displaystyle \text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}\)

Plug in the lengths of the half diagonals to find the length of the rhombus.

\(\displaystyle \text{Side of Rhombus}=\sqrt{26^2+29^2}=\sqrt{1517}=38.95\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

Example Question #35 : How To Find The Length Of The Side Of A Rhombus

Find the length of a side of a rhombus that has diagonals with lengths of \(\displaystyle 58\) and \(\displaystyle 60\).

Possible Answers:

\(\displaystyle 49.90\)

\(\displaystyle 41.73\)

\(\displaystyle 42.30\)

\(\displaystyle 39.01\)

Correct answer:

\(\displaystyle 41.73\)

Explanation:

13

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

\(\displaystyle \text{Half Diagonal 1}=\frac{58}{2}=29\)

\(\displaystyle \text{Half Diagonal 2}=\frac{60}{2}=30\)

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

\(\displaystyle \text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}\)

Plug in the lengths of the half diagonals to find the length of the rhombus.

\(\displaystyle \text{Side of Rhombus}=\sqrt{29^2+30^2}=\sqrt{1741}=41.73\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

Example Question #36 : How To Find The Length Of The Side Of A Rhombus

Find the length of a side of a rhombus that has diagonals with side lengths of \(\displaystyle 100\) and \(\displaystyle 108\).

Possible Answers:

\(\displaystyle 73.59\)

\(\displaystyle 77.72\)

\(\displaystyle 76.88\)

\(\displaystyle 74.51\)

Correct answer:

\(\displaystyle 73.59\)

Explanation:

13

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

\(\displaystyle \text{Half Diagonal 1}=\frac{100}{2}=50\)

\(\displaystyle \text{Half Diagonal 2}=\frac{108}{2}=54\)

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

\(\displaystyle \text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}\)

Plug in the lengths of the half diagonals to find the length of the rhombus.

\(\displaystyle \text{Side of Rhombus}=\sqrt{50^2+54^2}=\sqrt{5416}=73.59\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

Example Question #31 : How To Find The Length Of The Side Of A Rhombus

Given: Parallelogram \(\displaystyle ABCD\) such that \(\displaystyle AB = CD= 12\).

True or false: Parallelogram \(\displaystyle ABCD\) must be a rhombus.

Possible Answers:

True

False

Correct answer:

False

Explanation:

A rhombus is defined to be a quadrilateral with four congruent sides. 

Parallelogram \(\displaystyle ABCD\) gives the lengths of two of its opposite sides to be congruent, but this is characteristic of all parallelograms. No information is given about the other two sides, so the figure need not be a rhombus.

Example Question #251 : Intermediate Geometry

Given: Parallelogram \(\displaystyle ABCD\) such that \(\displaystyle AB = BC= 12\).

True or false: Parallelogram \(\displaystyle ABCD\) must be a rhombus.

Possible Answers:

False

True

Correct answer:

True

Explanation:

Opposite sides of a parallelogram are congruent, so 

\(\displaystyle CD = AB = 12\)

and 

\(\displaystyle AD = BC = 12\)

All four sides are congruent to one another. It follows by definition that Parallelogram \(\displaystyle ABCD\) is a rhombus.

Example Question #31 : How To Find The Length Of The Side Of A Rhombus

Given: Quadrilateral \(\displaystyle ABCD\) with diagonal \(\displaystyle \overline{AC}\); \(\displaystyle \bigtriangleup ABC \cong \bigtriangleup ADC\).

True or false: From the information given, it follows that Quadrilateral \(\displaystyle ABCD\) is a parallelogram.

Possible Answers:

False

True

Correct answer:

False

Explanation:

Below are two quadrilaterals marked \(\displaystyle ABCD\) with \(\displaystyle \overline{AC}\) drawn.

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The quadrilateral on the left has four congruent sides and is by definition a rhombus. The quadrilateral on the right is not a rhombus, since not all four sides are congruent.

In both cases, \(\displaystyle \overline{AB} \cong \overline{AD}\), \(\displaystyle \overline{CB} \cong \overline{CD}\), and, by the reflexive property, \(\displaystyle \overline{AC} \cong \overline{AC}\). By the Side-Side-Side Congruence Postulate, it can be proved that \(\displaystyle \bigtriangleup ABC \cong \bigtriangleup ADC\) in both diagrams.

Therefore, Quadrilateral \(\displaystyle ABCD\) need not be a rhombus.

Example Question #40 : How To Find The Length Of The Side Of A Rhombus

Given: \(\displaystyle \bigtriangleup ABC\) and \(\displaystyle \bigtriangleup DEF\).

\(\displaystyle \overline{AB} \cong \overline{DE}\)

\(\displaystyle \overline{AC} \cong \overline{DF}\)

\(\displaystyle \angle C \cong \angle E\)

True or false: It follows from the given information that \(\displaystyle \bigtriangleup ABC \cong \bigtriangleup DEF\).

Possible Answers:

True

False

Correct answer:

False

Explanation:

As we are establishing whether or not \(\displaystyle \bigtriangleup ABC \cong \bigtriangleup DEF\), then \(\displaystyle A\)\(\displaystyle B\), and \(\displaystyle C\) correspond respectively to \(\displaystyle D\)\(\displaystyle E\), and \(\displaystyle F\).

If we examine the sides and the angle of \(\displaystyle \bigtriangleup ABC\) in the congruence statements - \(\displaystyle \overline{AB}\), \(\displaystyle \overline{AC}\), and \(\displaystyle \angle C\) - we see that these are two sides and a nonincluded angle. But if we examine the sides and angles of \(\displaystyle \bigtriangleup DEF\) -\(\displaystyle \overline{DE}\)\(\displaystyle \overline{DF}\), and \(\displaystyle \angle E\) - we see that these are two sides and an included angle.

The only congruence postulate dealing with two sides and an angle is the SAS Congruence Postulate, which requires congruence between two sides and the included angle of both triangles. This theorem does not apply, so we cannot prove the triangles congruent.

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