Sure, let's break down each part of the question step-by-step.
### 1. Finding the Height of a Triangle
We are given:
- Area of the triangle, [tex]\( A = 27 \, \text{ft}^2 \)[/tex]
- Base of the triangle, [tex]\( b = 9 \, \text{ft} \)[/tex]
The formula for the area of a triangle is:
[tex]\[ A = \frac{1}{2} b \cdot h \][/tex]
To find the height ([tex]\( h \)[/tex]), we can rearrange the formula:
[tex]\[ h = \frac{2A}{b} \][/tex]
Substitute the given values:
[tex]\[ h = \frac{2 \cdot 27}{9} = \frac{54}{9} = 6 \, \text{ft} \][/tex]
So, the height of the triangle is [tex]\( 6 \, \text{ft} \)[/tex].
The choices were:
- [tex]\( 243 \, \text{ft} \)[/tex]
- [tex]\( 6 \, \text{ft} \)[/tex] (Correct choice)
- [tex]\( 121.5 \, \text{ft} \)[/tex]
### 2. Finding the Area of a Trapezoid
We are given:
- First base of the trapezoid, [tex]\( b_1 = 4 \, \text{cm} \)[/tex]
- Second base of the trapezoid, [tex]\( b_2 = 5 \, \text{cm} \)[/tex]
- Height of the trapezoid, [tex]\( h = 3 \, \text{cm} \)[/tex]
The formula for the area of a trapezoid is:
[tex]\[ A = \frac{1}{2} (b_1 + b_2) \cdot h \][/tex]
Substitute the given values:
[tex]\[ A = \frac{1}{2} (4 + 5) \cdot 3 = \frac{1}{2} \cdot 9 \cdot 3 = \frac{27}{2} = 13.5 \, \text{cm}^2 \][/tex]
So, the area of the trapezoid is [tex]\( 13.5 \, \text{cm}^2 \)[/tex].
The choices were:
- [tex]\( 13.5 \, \text{cm}^2 \)[/tex] (Correct choice)
- [tex]\( 30 \, \text{cm}^2 \)[/tex]
- [tex]\( 27 \, \text{cm}^2 \)[/tex]
- [tex]\( 6 \, \text{cm}^2 \)[/tex]
### 3. Finding the Height of a Trapezoid
We are given:
- First base of the trapezoid, [tex]\( b_1 = 9 \, \text{yd} \)[/tex]
- Second base of the trapezoid, [tex]\( b_2 = 12 \, \text{yd} \)[/tex]
- Area of the trapezoid, [tex]\( A = 21 \, \text{square yards} \)[/tex]
The formula for the height of a trapezoid is:
[tex]\[ h = \frac{2A}{b_1 + b_2} \][/tex]
Substitute the given values:
[tex]\[ h = \frac{2 \cdot 21}{9 + 12} = \frac{42}{21} = 2 \, \text{yd} \][/tex]
So, the height of the trapezoid is [tex]\( 2 \, \text{yd} \)[/tex].
The choices were:
- [tex]\( 5 \, \text{yd} \)[/tex]
- [tex]\( 220.5 \, \text{yd} \)[/tex]
- [tex]\( 1 \, \text{yd} \)[/tex]
- [tex]\( 2 \, \text{yd} \)[/tex] (Correct choice)
To summarize:
1. The height of the triangle is [tex]\( 6 \, \text{ft} \)[/tex].
2. The area of the trapezoid is [tex]\( 13.5 \, \text{cm}^2 \)[/tex].
3. The height of the trapezoid is [tex]\( 2 \, \text{yd} \)[/tex].
