section 2: special right triangles (10 problems - calculators ok)
9. 45-45-90 triangle: hypotenuse = 20. legs = ______ each
10. 45-45-90 triangle: each leg = 11. hypotenuse = ______
11. 30-60-90 triangle: shorter leg = 7. longer leg = ______ hypotenuse = ______
12. 30-60-90 triangle: longer leg = 10√3. shorter leg = ______ hypotenuse = ______
13. 30-60-90 triangle: hypotenuse = 24. shorter leg = ______ longer leg = ______
14. a 20-meter ladder leans against a wall at 45°. how far up the wall does it reach?
15. tree casts 18 m shadow, sun makes 30° angle of elevation with ground. tree height = ______
16. equilateral triangle has side length 16. find the exact area.

Question

Accepted Answer

### Problem 9:
# Explanation:
## Step1: Recall 45-45-90 triangle ratios
In a 45-45-90 triangle, the legs are equal, and the hypotenuse $ h $ is related to the leg $ l $ by $ h = l\sqrt{2} $, so $ l=\frac{h}{\sqrt{2}} $.
## Step2: Substitute hypotenuse value
Given $ h = 20 $, then $ l=\frac{20}{\sqrt{2}}=\frac{20\sqrt{2}}{2}=10\sqrt{2} $.
# Answer: $ 10\sqrt{2} $

### Problem 10:
# Explanation:
## Step1: Recall 45-45-90 triangle ratios
For a 45-45-90 triangle, hypotenuse $ h = l\sqrt{2} $, where $ l $ is the leg length.
## Step2: Substitute leg value
Given $ l = 11 $, so $ h = 11\sqrt{2} $.
# Answer: $ 11\sqrt{2} $

### Problem 11:
# Explanation:
## Step1: Recall 30-60-90 triangle ratios
In a 30-60-90 triangle, shorter leg $ s $, longer leg $ l = s\sqrt{3} $, hypotenuse $ h = 2s $.
## Step2: Find longer leg
Given $ s = 7 $, longer leg $ l = 7\sqrt{3} $.
## Step3: Find hypotenuse
Hypotenuse $ h = 2\times7 = 14 $.
# Answer: Longer leg: $ 7\sqrt{3} $, Hypotenuse: $ 14 $

### Problem 12:
# Explanation:
## Step1: Recall 30-60-90 triangle ratios
Longer leg $ l = s\sqrt{3} $, so $ s=\frac{l}{\sqrt{3}} $, hypotenuse $ h = 2s $.
## Step2: Find shorter leg
Given $ l = 10\sqrt{3} $, $ s=\frac{10\sqrt{3}}{\sqrt{3}} = 10 $.
## Step3: Find hypotenuse
Hypotenuse $ h = 2\times10 = 20 $.
# Answer: Shorter leg: $ 10 $, Hypotenuse: $ 20 $

### Problem 13:
# Explanation:
## Step1: Recall 30-60-90 triangle ratios
Hypotenuse $ h = 2s $, so shorter leg $ s=\frac{h}{2} $, longer leg $ l = s\sqrt{3} $.
## Step2: Find shorter leg
Given $ h = 24 $, $ s=\frac{24}{2}=12 $.
## Step3: Find longer leg
Longer leg $ l = 12\sqrt{3} $.
# Answer: Shorter leg: $ 12 $, Longer leg: $ 12\sqrt{3} $

### Problem 14:
# Explanation:
## Step1: Model the situation as a 45-45-90 triangle
The ladder is the hypotenuse, and the height on the wall is a leg (since angle with wall is $ 45^\circ $, so it's a 45-45-90 triangle).
## Step2: Use 45-45-90 triangle ratio
Let height be $ h $, hypotenuse (ladder) $ = 20 $ m. From 45-45-90 ratio, $ h=\frac{20}{\sqrt{2}} = 10\sqrt{2} \approx14.14 $ m (or exact form $ 10\sqrt{2} $).
# Answer: $ 10\sqrt{2} $ meters (or $ \approx14.14 $ meters)

### Problem 15:
# Explanation:
## Step1: Model the situation as a 30-60-90 triangle
Shadow length is the adjacent side (longer leg or shorter leg? Sun angle $ 30^\circ $, so tree height is opposite side (shorter leg? Wait, angle of elevation $ 30^\circ $, so tree height $ h $, shadow length $ 18 $ m (adjacent side, longer leg in 30-60-90? Wait, no: for angle of elevation $ 30^\circ $, $ \tan(30^\circ)=\frac{h}{18} $, but also, in 30-60-90, if shadow is adjacent (longer leg), then $ h = \frac{18}{\sqrt{3}}=6\sqrt{3} $ (since $ \tan(30^\circ)=\frac{1}{\sqrt{3}}=\frac{h}{18}\Rightarrow h = \frac{18}{\sqrt{3}} = 6\sqrt{3} \approx10.39 $ m).
## Step2: Calculate tree height
Using $ \tan(30^\circ)=\frac{h}{18} $, $ h = 18\times\frac{1}{\sqrt{3}} = 6\sqrt{3} $.
# Answer: $ 6\sqrt{3} $ meters (or $ \approx10.39 $ meters)

### Problem 16:
# Explanation:
## Step1: Recall area of equilateral triangle formula
Area of equilateral triangle with side $ a $ is $ A=\frac{\sqrt{3}}{4}a^2 $.
## Step2: Substitute side length
Given $ a = 16 $, so $ A=\frac{\sqrt{3}}{4}\times16^2=\frac{\sqrt{3}}{4}\times256 = 64\sqrt{3} $.
# Answer: $ 64\sqrt{3} $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 9:

Step1: Recall 45-45-90 triangle ratios

Step2: Substitute hypotenuse value

Step1: Recall 45-45-90 triangle ratios

Step2: Substitute leg value

Step1: Recall 30-60-90 triangle ratios

Step2: Find longer leg

Step3: Find hypotenuse

Answer:

Problem 10: