Question

30°-60°-90° triangle theorem: equilateral triangles
in a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

• sketch a perpendicular bisector ad; that means were going to bisect ∠bac.
• use the pythagorean theorem.

x² + (ad)² = (______)²
x² + (ad)² = 4x²
(ad)² = ______²
______ = x√3

• ____ : x√3 : ____
• 1 : √3 : ______
• so when we generalize these ratios to any 30°-60°-90° triangle, we note the connection of a similarity, which is that all 30°-60°-90° triangles are similar because the angles are __________.
• since sides of similar figures are proportional we know this relationship will hold.

(image of triangle abc with ad as perpendicular bisector from a to bc, labeled with a at top, b and c at base, d midpoint of bc, ab labeled 2x, angle at a (∠bad) labeled 30°)

Explanation:

Step1: Identify the hypotenuse

In the right triangle \(ABD\), the hypotenuse is \(AB\), which has length \(2x\) (from the diagram). So the first blank is \(2x\), and the equation is \(x^{2}+(AD)^{2}=(2x)^{2}\).

Step2: Solve for \((AD)^{2}\)

Subtract \(x^{2}\) from both sides: \((AD)^{2}=4x^{2}-x^{2}=3x^{2}\).

Step3: Solve for \(AD\)

Take the square root of both sides: \(AD = \sqrt{3x^{2}}=x\sqrt{3}\) (since length is positive).

Step4: Side ratios of \(30^\circ - 60^\circ - 90^\circ\) triangle

The sides are \(x\) (shorter leg), \(x\sqrt{3}\) (longer leg), and \(2x\) (hypotenuse). So the ratio is \(x:x\sqrt{3}:2x\), which simplifies to \(1:\sqrt{3}:2\) when divided by \(x\).

Step5: Similar triangles reason

All \(30^\circ - 60^\circ - 90^\circ\) triangles are similar because their angles are equal (AA similarity criterion: \(30^\circ\), \(60^\circ\), and \(90^\circ\) angles).

Answer:

• First blank (hypotenuse): \(2x\)
• \((AD)^{2}\): \(3x^{2}\)
• \(AD\): \(x\sqrt{3}\)
• Side ratio: \(x:x\sqrt{3}:2x\) (or simplified \(1:\sqrt{3}:2\))
• Angles reason: equal (or congruent)