Question

30° 60° 90° special right triangles

1. find and label the measure of each missing angle for each triangle.
2. how many times bigger is the hypotenuse than the short leg for each triangle?
3. find the long leg for each triangle above using the pythagorean theorem. add your answers to the triangle diagrams in simplest radical form.
4. how many times bigger is the long leg than the short leg?

Explanation:

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Question 1: Find Missing Angles

Step1: Triangle ABC: Sum angles to 180°

$\angle C = 180^\circ - 90^\circ - 30^\circ = 60^\circ$

Step2: Triangle XYZ: Sum angles to 180°

$\angle Z = 180^\circ - 90^\circ - 30^\circ = 60^\circ$

Step3: Triangle QRS: Sum angles to 180°

$\angle Q = 180^\circ - 90^\circ - 60^\circ = 30^\circ$

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Question 2: Hypotenuse vs Short Leg

Step1: Triangle ABC: Identify sides

Short leg $BC=4$, Hypotenuse $AC=8$
$\frac{8}{4}=2$

Step2: Triangle XYZ: Identify sides

Short leg $YZ=3$, Hypotenuse $XZ=6$
$\frac{6}{3}=2$

Step3: Triangle QRS: Identify sides

Short leg $RS=2$, Hypotenuse $QS=4$
$\frac{4}{2}=2$

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Question 3: Find Long Leg (Pythagorean Theorem: $a^2 + b^2 = c^2$, rearranged to $b=\sqrt{c^2 - a^2}$)

Step1: Triangle ABC: Calculate long leg AB

$AB = \sqrt{8^2 - 4^2} = \sqrt{64-16} = \sqrt{48} = 4\sqrt{3}$

Step2: Triangle XYZ: Calculate long leg XY

$XY = \sqrt{6^2 - 3^2} = \sqrt{36-9} = \sqrt{27} = 3\sqrt{3}$

Step3: Triangle QRS: Calculate long leg QR

$QR = \sqrt{4^2 - 2^2} = \sqrt{16-4} = \sqrt{12} = 2\sqrt{3}$

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Question 4: Long Leg vs Short Leg

Step1: Triangle ABC: Compare legs

$\frac{4\sqrt{3}}{4}=\sqrt{3}$

Step2: Triangle XYZ: Compare legs

$\frac{3\sqrt{3}}{3}=\sqrt{3}$

Step3: Triangle QRS: Compare legs

$\frac{2\sqrt{3}}{2}=\sqrt{3}$

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Answer:

1. Missing angles:

• Triangle ABC: $\angle C = 60^\circ$
• Triangle XYZ: $\angle Z = 60^\circ$
• Triangle QRS: $\angle Q = 30^\circ$

1. For all triangles, the hypotenuse is 2 times bigger than the short leg.
2. Long legs (simplest radical form):

• Triangle ABC: $AB = 4\sqrt{3}$
• Triangle XYZ: $XY = 3\sqrt{3}$
• Triangle QRS: $QR = 2\sqrt{3}$

1. For all triangles, the long leg is $\boldsymbol{\sqrt{3}}$ times bigger than the short leg.