Question

the area of a rhombus is 540 cm² and the length of one of its diagonals is 4.5 dm. what is the distance from the point of intersection of the diagonals to a side of the rhombus? answer: the distance is cm.

Explanation:

Step1: Convert diagonal length unit

$4.5\ dm = 45\ cm$

Step2: Find the other diagonal

Let the other diagonal be $d_2$. Using area formula $A=\frac{1}{2}d_1d_2$, we have $540=\frac{1}{2}\times45\times d_2$, so $d_2 = 24\ cm$.

Step3: Find half - diagonals

Half - diagonals are $a = \frac{45}{2}=22.5\ cm$ and $b = 12\ cm$.

Step4: Find side length of rhombus

Side length $s=\sqrt{22.5^{2}+12^{2}}=\sqrt{506.25 + 144}=\sqrt{650.25}=25.5\ cm$.

Step5: Find the distance

Let the distance be $h$. Using area of small right - triangle formed, $\frac{1}{2}\times s\times h=\frac{1}{4}\times540$. Substituting $s = 25.5\ cm$, we get $h=\frac{135}{25.5}=\frac{1350}{255}=\frac{90}{17}\approx5.29\ cm$.

Answer:

$\frac{90}{17}$