Question

30°-60°-90° theorem
complete the sentence:
in a 30°-60°-90° triangle, the hypotenuse is _________ the short leg
in a 30°-60°-90° triangle, the long leg is _________ times the short leg
45°-45°-90° theorem
in a 45°-45°-90° triangle, the hypotenuse is _________ times the leg.
in a 45°-45°-90° triangle, both legs are _________.
fill in the missing values on the triangles.

if the hypotenuse in a 30°-60°-90° is equal to 12 inches, find the short leg and the long leg.
short leg = _______ long leg = _______
if the hypotenuse in a 45°-45°-90° is equal to 12 inches, what is the length of the legs?
leg = _________
what is the decimal approximation for $sqrt{2}$? _________
round your answer to the nearest thousandth.
what is the decimal approximation for $sqrt{3}$? _________
round your answer to the nearest thousandth.

Explanation:

Step1: 30-60-90 triangle hypotenuse rule

In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, hypotenuse $= 2 \times$ short leg.

Step2: 30-60-90 triangle long leg rule

In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, long leg $= \sqrt{3} \times$ short leg.

Step3: 45-45-90 triangle hypotenuse rule

In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, hypotenuse $= \sqrt{2} \times$ leg.

Step4: 45-45-90 triangle leg rule

In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, both legs are congruent (equal).

Step5: Calculate 30-60-90 short leg

Given hypotenuse $=12$, short leg $= \frac{12}{2} = 6$ inches.

Step6: Calculate 30-60-90 long leg

Long leg $= 6 \times \sqrt{3}$ inches.

Step7: Calculate 45-45-90 leg length

Given hypotenuse $=12$, leg $= \frac{12}{\sqrt{2}} = 6\sqrt{2}$ inches.

Step8: Approximate $\sqrt{2}$

$\sqrt{2} \approx 1.414$ (nearest thousandth)

Step9: Approximate $\sqrt{3}$

$\sqrt{3} \approx 1.732$ (nearest thousandth)

Answer:

Completed Sentences:

1. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, the hypotenuse is 2 times the short leg.
2. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, the long leg is $\boldsymbol{\sqrt{3}}$ times the short leg.
3. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, the hypotenuse is $\boldsymbol{\sqrt{2}}$ times the leg.
4. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, both legs are equal (congruent).

Calculations:

• Short leg (30-60-90 triangle) = $\boldsymbol{6}$ inches
• Long leg (30-60-90 triangle) = $\boldsymbol{6\sqrt{3}}$ (or ~10.392) inches
• Leg (45-45-90 triangle) = $\boldsymbol{6\sqrt{3}}$ (or ~8.485) inches
• Decimal approximation of $\sqrt{2}$ = $\boldsymbol{1.414}$
• Decimal approximation of $\sqrt{3}$ = $\boldsymbol{1.732}$