The diagonals of the rectangle ABCD meet at point O. The segment ОF is the height of the triangle AOD.

The diagonals of the rectangle ABCD meet at point O. The segment ОF is the height of the triangle AOD. Calculate the degree measures of the acute angles of the triangle AOF if the area of the rectangle is 16√3 cm2 and AD = 4 cm.

Let’s write the formula for the area of a rectangle:
S = a * b. In our case:
S = AD * CD → CD = S / AD = 16√3 / 4 = 4√3 (cm).
By the Pythagorean theorem, we find the diagonal AC:
AC = √ (AD² + CD²) = √ (16 + 48) = √ 64 = 8 (cm).
Point O divides the diagonals in half, AO = 1/2 * AC = 4 (cm).
In an isosceles triangle AOD, the height OF is also a median and a bisector, therefore:
AF = 1/2 * AD = 2 (cm).
Consider a right-angled triangle AOF, in which the hypotenuse AO = 4 cm is known, and the leg AF = 2 cm. The leg is equal to half of the hypotenuse, which means that the angle AOF = 30 °. Second acute angle OAF = 90 ° – 30 ° = 60 °.
Answer: The acute angles of the triangle AOF are 30 ° and 60 °.