for this activity, you must help bob, the special right triangle turkey, color his feathers! be aware that each feather needs a specific color depending on the special right triangle it is. solve for the triangle measurements and color bob the special right triangle turkey! *** if the triangle is 30 - 60 - 90, color them orange! *** *** if the triangle is 45 - 45 - 90, color them yellow! *** 1) 2) 3) 4) 5) 6)

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Accepted Answer

### Problem 1:
# Explanation:
## Step1: Identify triangle type
It's a 45-45-90 triangle (right angle and 45° angle, so the other angle is 45°). In a 45-45-90 triangle, legs are equal, and hypotenuse $ a = \text{leg} \times \sqrt{2} $, legs are equal. Given one leg is $ 2\sqrt{2} $, so $ b = 2\sqrt{2} $ (since legs are equal in 45-45-90), and hypotenuse $ a = 2\sqrt{2} \times \sqrt{2} = 4 $.
## Step2: Determine color
Since it's 45-45-90, color is Yellow.
# Answer: $ b = 2\sqrt{2} $, $ a = 4 $, Color: Yellow

### Problem 2:
# Explanation:
## Step1: Identify triangle type
45-45-90 triangle (right angle and 45° angle). Hypotenuse is 4. In 45-45-90, $ \text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}} $. So $ x = y = \frac{4}{\sqrt{2}} = 2\sqrt{2} $ (rationalizing the denominator: $ \frac{4\sqrt{2}}{2} = 2\sqrt{2} $).
## Step2: Determine color
45-45-90, so color Yellow.
# Answer: $ x = 2\sqrt{2} $, $ y = 2\sqrt{2} $, Color: Yellow

### Problem 3:
# Explanation:
## Step1: Identify triangle type
30-60-90 triangle (right angle and 60° angle, so the other angle is 30°). In 30-60-90, the sides are in ratio $ 1 : \sqrt{3} : 2 $ (opposite 30°, 60°, 90° respectively). The side opposite 60° is $ 5\sqrt{3} $, so the side opposite 30° (y) is $ \frac{5\sqrt{3}}{\sqrt{3}} = 5 $, and hypotenuse (x) is $ 5 \times 2 = 10 $.
## Step2: Determine color
30-60-90, so color Orange.
# Answer: $ y = 5 $, $ x = 10 $, Color: Orange

### Problem 4:
# Explanation:
## Step1: Analyze triangle
Right triangle with hypotenuse $ 8\sqrt{5} $. But we need more info? Wait, maybe it's a 45-45-90? No, if legs are equal, hypotenuse $ = \text{leg} \times \sqrt{2} $, but $ 8\sqrt{5} $ doesn't fit. Wait, maybe 30-60-90? But no angles given. Wait, maybe the problem has a typo or missing info? Wait, the triangle is right-angled, with hypotenuse $ 8\sqrt{5} $. If we assume it's 45-45-90, legs $ x = y = \frac{8\sqrt{5}}{\sqrt{2}} = 4\sqrt{10} $. But without angle, hard to tell. Wait, maybe the original problem has angles? Wait, the diagram shows a right angle, but no other angle. Wait, maybe it's a different case. Wait, maybe the triangle is isoceles right triangle (45-45-90), so legs equal. So $ x = y = \frac{8\sqrt{5}}{\sqrt{2}} = 4\sqrt{10} $, color Yellow (if 45-45-90). But maybe I misread. Wait, the problem says "Special Right Triangle", so either 30-60-90 or 45-45-90. Since hypotenuse is $ 8\sqrt{5} $, if it's 45-45-90, legs are $ 4\sqrt{10} $. So:
## Step1: Calculate legs
$ x = y = \frac{8\sqrt{5}}{\sqrt{2}} = 4\sqrt{10} $ (rationalizing: $ \frac{8\sqrt{5}\times\sqrt{2}}{2} = 4\sqrt{10} $).
## Step2: Determine color (assuming 45-45-90)
Color Yellow.
# Answer: $ x = 4\sqrt{10} $, $ y = 4\sqrt{10} $, Color: Yellow (assuming 45-45-90, if angle was 45°; if not, more info needed, but as special right triangle, likely 45-45-90 or 30-60-90. Since hypotenuse doesn't fit 30-60-90 ratio with integer/radical sides, 45-45-90 is more likely.)

### Problem 5:
# Explanation:
## Step1: Analyze triangle
Right triangle with hypotenuse 3. But we need to know angles or other sides. Wait, maybe it's 30-60-90? If hypotenuse is 3, then side opposite 30° is $ \frac{3}{2} = 1.5 $, side opposite 60° is $ \frac{3\sqrt{3}}{2} $. But no angles given. Wait, maybe the triangle is 45-45-90? Then legs $ a = b = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} $. But without angle, hard to tell. Wait, the problem says "Special Right Triangle", so either 30-60-90 or 45-45-90. Since hypotenuse is 3, if 45-45-90, legs $ \frac{3\sqrt{2}}{2} $; if 30-60-90, legs $ 1.5 $ and $ \frac{3\sqrt{3}}{2} $. But maybe the diagram has angles? Wait, the diagram shows a right angle, but no other angle. So maybe missing info. But assuming it's 45-45-90 (isoceles right triangle), then $ a = b = \frac{3\sqrt{2}}{2} $, color Yellow. If 30-60-90, color Orange. But since no angle, maybe the problem expects us to assume? Wait, maybe the original problem has angles. Alternatively, maybe it's a 3-4-5 triangle? No, hypotenuse is 3, so legs would be less. Wait, maybe I made a mistake. Wait, the hypotenuse is 3, so by Pythagoras, $ a^2 + b^2 = 9 $. If it's 45-45-90, $ 2a^2 = 9 \implies a = b = \frac{3\sqrt{2}}{2} $. So:
## Step1: Calculate sides (assuming 45-45-90)
$ a = b = \frac{3\sqrt{2}}{2} $
## Step2: Determine color (assuming 45-45-90)
Color Yellow.
# Answer: $ a = b = \frac{3\sqrt{2}}{2} $ (assuming 45-45-90), Color: Yellow (or Orange if 30-60-90, but more likely 45-45-90 if isoceles)

### Problem 6:
# Explanation:
## Step1: Identify triangle type
45-45-90 triangle (right angle and legs marked equal, so angles are 45-45-90). Hypotenuse is $ 8\sqrt{2} $. In 45-45-90, $ \text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}} $. So $ u = v = \frac{8\sqrt{2}}{\sqrt{2}} = 8 $.
## Step2: Determine color
45-45-90, so color Yellow.
# Answer: $ u = 8 $, $ v = 8 $, Color: Yellow

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Explanation:

Problem 1:

Step1: Identify triangle type

Step2: Determine color

Step1: Identify triangle type

Step2: Determine color

Step1: Identify triangle type

Step2: Determine color

Answer:

Problem 2: