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highshill · Jul 9, 2017

A diagonal of a rhombus with the size 9 cm makes a angle of 16 degrees with the side Find the length of the diagonalso

bujolover · Jul 9, 2017

highshill said:
A diagonal of a rhombus with the size 9 cm makes a angle of 16 degrees with the side Find the length of the diagonalso

Let the length of diagonal AC be x cm.

AD = CD = 9cm (equal sides of rhombus ABCD)

∠DCB = 2*16° = 32° (diagonal AC bisects vertex ∠DCB through which it passes)

AD||BC (opposite sides of rhombus ABCD parallel)
Therefore, ∠D + ∠DCB = 180° (co-interior angles supplementary on AD||BC)
Therefore, ∠D = 180° - 32° = 148°

Now, x² = 9² + 9² - 2*9*9cos148° (cosine rule)
x² = 81*2 - 162cos148°
x² = 299.3837916

x = 17.30271053cm
Therefore, x = 17.30cm (2 dec. pl.)

1729 · Jul 9, 2017

highshill said:
A diagonal of a rhombus with the size 9 cm makes a angle of 16 degrees with the side Find the length of the diagonalso

$$\noindent Use the property that the diagonals of a rhombus bisect each other at right angles. Also, the diagonals of a rhombus divide the rhombus into four congruent triangles. \\\\ In any one of these triangles, if $l$ is the length of the other diagonal, \\ \begin{align*} \quad \tan{16\degree} &= \frac{\left(\frac{l}{2}\right)}{4.5} \\ l &= 2 \times 4.5\tan{16\degree} \\ &\approx 2.58 \text{ cm} \end{align*} \\ Therefore the length of the diagonals are 9 cm and 2.58 cm.$

1729 · Jul 9, 2017

bujolover · Jul 9, 2017

1729 said:
$$\noindent My interpretation of the question meant that the diagonal itself was 9 cm rather than the side, and the length of the other diagonal was being asked for. \\\\ If the question meant the rhombus had \textbf{side 9 cm} then it simply becomes, \\ \begin{align*}\quad \cos{16\degree} &= \frac{\left(\frac{l}{2}\right)}{9} \\ l &= 2 \times 9\cos{16\degree} \\ &\approx 17.3 \text{ cm} \end{align*} \\ The length of the other diagonal can be found using Pythagoras' Theorem.$

I did it the long way, lol.

pikachu975 · Jul 9, 2017

bujolover said:
I did it the long way, lol.

The diagonal is 9 cm not the side I think

highshill · Jul 10, 2017

that was confusing me as well.

bujolover · Jul 11, 2017

highshill said:
that was confusing me as well.

I'm pretty sure it's talking about the side being 9cm, otherwise it wouldn't make much sense to find the "diagonals" when one is already given. If the diagonal length is 9cm, then it should ask you to find the "other diagonal", not the "diagonals". Check the answer to see if you can tell which one is 9cm.

pikachu975 · Jul 11, 2017

bujolover said:
I'm pretty sure it's talking about the side being 9cm, otherwise it wouldn't make much sense to find the "diagonals" when one is already given. If the diagonal length is 9cm, then it should ask you to find the "other diagonal", not the "diagonals". Check the answer to see if you can tell which one is 9cm.

A diagonal of a rhombus with the size 9 cm makes a angle of 16 degrees with the side Find the length of the diagonals

I think it's talking about the diagonal because it's saying the diagonal makes an angle of 16 degrees with the side so the 9 cm should be talking about the diagonal still. But yeah check the answers just in case.

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