Answer:
In a rectangle, the diagonals bisect each other, forming four right-angled triangles. Given the sides of the rectangle, you can use the Pythagorean theorem to find the length of the diagonals. The acute angle between the diagonals can then be determined using trigonometric functions.
Let \(a\) and \(b\) be the sides of the rectangle (length and width). The length of the diagonals (\(d\)) can be found using the Pythagorean theorem:
\[ d = \sqrt{a^2 + b^2} \]
For a rectangle with sides 5 cm and 7 cm:
\[ d = \sqrt{5^2 + 7^2} \]
\[ d = \sqrt{25 + 49} \]
\[ d = \sqrt{74} \]
Now, you can find the acute angle (\(\theta\)) between the diagonals using trigonometric functions. In a rectangle, the diagonals bisect each other, so each acute angle (\(\theta\)) is half of the angle formed by the diagonals.
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
\[ \tan(\theta) = \frac{\frac{b}{2}}{\frac{a}{2}} \]
\[ \tan(\theta) = \frac{b}{a} \]
Substitute the values of \(a\) and \(b\) from the rectangle:
\[ \tan(\theta) = \frac{7}{5} \]
Now, find the acute angle (\(\theta\)) by taking the arctan (inverse tangent) of \(\frac{7}{5}\).
\[ \theta = \arctan\left(\frac{7}{5}\right) \]
Calculate this value to find the acute angle between the diagonals of the rectangle.
