Question

ABCD is a rectangle. Find y and ∠ABD.

Answer 1

Given:

• ABCD is a rectangle.
• In the triangle ABD, ∠ADB = 3y and ∠DBA = 7y.
• We are to find y and ∠ABD.

Stepwise calculation:

1. Since ABCD is a rectangle, ∠ADC = 90°.

2. The sum of angles in triangle ABD is 180°. So, ∠ABD + ∠ADB + ∠DBA = 180°.

3. Substitute the given angles: ∠ABD + 3y + 7y = 180° ∠ABD + 10y = 180°.

4. Note that ∠ADC = 90° and diagonal BD splits the right angle at D into angles 3y (∠ADB) and 7y (∠BDC), so 3y + 7y = 90°, which gives 10y = 90° and therefore y = 9°.

5. Substitute y = 9° into equation from step 3: ∠ABD + 10 × 9° = 180°

∠ABD + 90° = 180°

∠ABD = 180° – 90°

∠ABD = 90°. 

However, from the figure, ∠ABD is given as the angle between AB and BD and the problem implies ∠ABD = 7y (not ∠DBA), so one must carefully identify the angles.

6. Considering the geometry carefully and ∠ABD = 27°, which means ∠ABD = 3y. 

Since ∠ADB = 3y and ∠DBA = 7y (angles given), their sum is still 10y = 90°, so y = 9°. 

Thus, ∠ABD = 3y = 3 × 9° = 27°.