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Advanced Geometry : Rhombuses

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Example Question #11 : How To Find The Length Of The Diagonal Of A Rhombus

Find the lengths of the two diagonals, the longer diagonal is $d_{1}$ , the shorter diagonal is $d_{2}$ .

Varsity1

Possible Answers:

$d_{1} = 2$

$d_{2} = 6$

$d_{1} = 6$

$d_{2} = 12$

$d_{1} = 6\cdot \sqrt{3}$

$d_{2} = 6$

$d_{1} = 13$

$d_{2} = 5$

$d_{1} = 12$

$d_{2} = 6$

Correct answer:

$d_{1} = 6\cdot \sqrt{3}$

$d_{2} = 6$

Explanation:

1) All sides of a rhombus are congruent.

2) Because all sides of a rhombus are congruent, the expressions of the side lengths can be set equal to each other. The resulting equation is then solved,

2x - 4 = 3x - 9

9 - 4 = 3x - 2x

5 = x

x = 5

3) Because the sides of a rhombus are congruent, x = 5 can be substituted into either 2x - 4 or 3x - 9 to find the length of a side,

2(5) - 4 = 6 , or, 3(5) - 9 = 6 .

4) Each of the composing triangles are right triangles, so then is the length of the hypotenuse for each triangle.

5) 6 = 3(2) .

6) The standard 30 - 60 - 90 right triangle has a hypotenuse length equal to .

7) The hypotenuse of a standard 30 - 60 - 90 right triangle is being multiplied by .

The result is , so then is the scale factor for the triangle side lengths.

8) For the standard 30 - 60 - 90 right triangle, the other two side lengths are and $\sqrt{3}$ , so then the height of the triangle from step 7) has a height of 3(1) , and the base length is $3(\sqrt{3})$ .

9) The base of the triangle from step 7) is

$\frac{1}{2}d_{1} = 3\cdot(\sqrt{3})$ ,

and the height is

$\frac{1}{2}d_{2} = 3(1)$ .

10) Diagonal

$d_2=2\cdot \frac{1}{2}d_{2} = 2\cdot3 = 6$ ,

and diagonal

$d_{1} = 2\cdot \frac{1}{2}d_{1} = 2\cdot3(\sqrt{3}) = 6(\sqrt{3})$ .

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Example Question #141 : Quadrilaterals

Screen_shot_2015-03-06_at_3.16.41_pm

What is the second diagonal for the above rhombus?

Possible Answers:

Correct answer:

Explanation:

Because a rhombus has vertical and horizontal symmetry, it can be broken into four congruent triangles, each with a hypotenuse of 13 and a base of 5 (half the given diagonal).

The Pythagorean Theorem

a^2+b^2=c^2 will yield,

$\sqrt{169-25}=12$ for the height of the triangles.

The greater diagonal is twice the height of the triangles therefore, the greater diagonal becomes:

$d=12 \cdot 2$

d=24

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Example Question #11 : How To Find The Length Of The Diagonal Of A Rhombus

ABCD is rhombus with side lengths in meters. $\overline{AB} = 13$ and $\overline{BD} = 10$ . What is the length, in meters, of $\overline{AC}$ ?

Rhombus_1

Possible Answers:

Correct answer:

Explanation:

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Rhombus_2

Thus, we can consider the right triangle AED to find the length of diagonal $\overline{AC}$ . From the given information, each of the sides of the rhombus measures meters and $\overline{BD} = 10$ .

Because the diagonals bisect each other, we know:

$\overline{AD} = 13$

$\overline{ED} = 5$

Using the Pythagorean theorem,

a^2 + b^2 = c^2

$\overline{AE}^2 + 5^2 = 13^2$

$\overline{AE}^2 = 144$

$\overline{AE} = 12$

$\overline{AC} = 24$

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Advanced Geometry : Rhombuses

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #11 : How To Find The Length Of The Diagonal Of A Rhombus

Example Question #141 : Quadrilaterals

Example Question #11 : How To Find The Length Of The Diagonal Of A Rhombus

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