Question

In rectangle ABCD, ∡BAC=30° and AD=50. What is the length of side DC?25.028.943.386.6

Answer 1

With the informationabout the rectangle, we can say the angle between the diagonal and the longest side (unknown) is 30°. While the shortest side is 50.

Then, to estimate the length of the remaining side (DC, which is equal to AB), we can state the following relationship:

$\tan 30^o=\frac{50}{AB}=\frac{50}{DC}$

Then:

$DC=\frac{50}{\tan30^o}=\frac{50}{\sqrt[]{3}/3}=\frac{3\cdot50}{\sqrt[]{3}}$

We can multiply numerator and denominator by square root of 3 to simplify:

$DC=\frac{3\cdot50}{\sqrt[]{3}}=\frac{\sqrt[]{3}\cdot3\cdot50}{\sqrt[]{3}\cdot\sqrt[]{3}}=\frac{\sqrt[]{3}\cdot3\cdot50}{3}=50\sqrt[]{3}\approx86.6$

The length of side DC is approximately 86.6. The correct answer is the fourth option.