The perpendicular, which is drawn from the top of the rectangle to its diagonal, divides the right angle in a ratio

The perpendicular, which is drawn from the top of the rectangle to its diagonal, divides the right angle in a ratio of 6: 3. Calculate the acute angle between the diagonals of the rectangle.

1. Vertices of the rectangle A, B, C, D. BH perpendicular to the diagonal BD. О – the point of intersection of the diagonals BD and AC.

2. By the condition of the problem ∠СBН: ∠ABН = 6: 3. That is, ∠СBН = 2∠ABН.

3.∠СBН + ∠ABН = 90 °. We replace ∠СBН in this expression with 2∠ABН:

∠ABН + 2∠ABН = 90 °.

∠ABH = 30 °.

4. ∠BAН = 180 ° – ∠ABH – ∠AНB = 180 ° – 30 ° – 90 ° = 60 °.

5. Triangle ABO – isosceles. Therefore, ∠ABO = ∠BAO = 60 °.

6. We calculate the acute angle between the diagonals ∠AOB:

∠AOB = 180 ° – (∠ABO + ∠BAO) = 180 ° – 120 ° = 60 °.

Answer: an acute angle between the diagonals ∠AOB = 60 °.