Question

verifying a 30°-60°-90° triangle relationship
given that △abc is equilateral, and ad bisects ∠a, show that the side lengths of △adb are in a ratio of 1 : √3 : 2.

1. equilateral triangles are also equiangular, so m∠a = m∠b = m∠c =

options: 30°, 45°, 60°

Explanation:

Step1: Sum angles in triangle

The sum of interior angles of a triangle is $180^\circ$. For an equilateral triangle, all angles are equal.
$$m\angle A = m\angle B = m\angle C = \frac{180^\circ}{3} = 60^\circ$$

Step2: Confirm angle bisector effect

AD bisects $\angle A$, so $\angle BAD = \frac{60^\circ}{2}=30^\circ$. AD is perpendicular to BC (in equilateral triangle, angle bisector = altitude), so $\angle ADB=90^\circ$.

Step3: Assign side length

Let $BD = x$. In $\triangle ADB$, $\angle BAD=30^\circ$, so hypotenuse $AB=2x$.

Step4: Calculate third side

Use Pythagorean theorem for $AD$:
$$AD = \sqrt{AB^2 - BD^2} = \sqrt{(2x)^2 - x^2} = \sqrt{3x^2} = x\sqrt{3}$$

Step5: Find side ratio

Ratio $BD:AD:AB = x:x\sqrt{3}:2x = 1:\sqrt{3}:2$

Answer:

1. $60^\circ$

The side lengths of $\triangle ADB$ are confirmed to be in the ratio $1:\sqrt{3}:2$.