Answer :
To find the equations that represent the relationship between the dimensions of the triangle, we need to use the formula for the area of a triangle:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
Given:
- Height of the triangle = 8 ft
- Area of the triangle = 42 ft²
Let's identify which equations are true based on this formula:
1. Equation A: [tex]\(42 = \frac{1}{2} \cdot b \cdot 8\)[/tex]
This equation matches the standard area formula. The equation is correct because it directly represents the relationship where the area is calculated using the base and height.
2. Equation D: [tex]\(4 \cdot b = 42\)[/tex]
If we solve for base [tex]\(b\)[/tex] using the area formula:
[tex]\[ 42 = \frac{1}{2} \cdot b \cdot 8 \][/tex]
Simplifying the equation:
[tex]\[ 42 = 4 \cdot b \][/tex]
Thus, this equation is also correct.
3. Equation B: [tex]\(42 \div w = \frac{1}{2} \cdot 8\)[/tex]
This equation implies an unknown [tex]\(w\)[/tex] and doesn't fit the format for calculating the area of a triangle using the base and height. Therefore, it is not correct.
4. Equation E: [tex]\(8 \cdot b = 42\)[/tex]
According to the area formula, multiplying base by height without the division by 2 doesn't relate correctly to the area. Thus, this is not valid.
5. Equation C: [tex]\(42 \div b = 8\)[/tex]
This implies dividing the area by the base equates the height, but doesn't match the operations needed for the area of a triangle in this problem. Hence, it's not accurate.
6. Equation F: [tex]\(b \div 8 = 42\)[/tex]
This equation suggests that dividing the base by the height equals the area, which doesn't correlate with the area formula for a triangle. Therefore, it is incorrect.
The correct equations for the given triangle's dimensions are:
- A. [tex]\(42 = \frac{1}{2} \cdot b \cdot 8\)[/tex]
- D. [tex]\(4 \cdot b = 42\)[/tex]
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
Given:
- Height of the triangle = 8 ft
- Area of the triangle = 42 ft²
Let's identify which equations are true based on this formula:
1. Equation A: [tex]\(42 = \frac{1}{2} \cdot b \cdot 8\)[/tex]
This equation matches the standard area formula. The equation is correct because it directly represents the relationship where the area is calculated using the base and height.
2. Equation D: [tex]\(4 \cdot b = 42\)[/tex]
If we solve for base [tex]\(b\)[/tex] using the area formula:
[tex]\[ 42 = \frac{1}{2} \cdot b \cdot 8 \][/tex]
Simplifying the equation:
[tex]\[ 42 = 4 \cdot b \][/tex]
Thus, this equation is also correct.
3. Equation B: [tex]\(42 \div w = \frac{1}{2} \cdot 8\)[/tex]
This equation implies an unknown [tex]\(w\)[/tex] and doesn't fit the format for calculating the area of a triangle using the base and height. Therefore, it is not correct.
4. Equation E: [tex]\(8 \cdot b = 42\)[/tex]
According to the area formula, multiplying base by height without the division by 2 doesn't relate correctly to the area. Thus, this is not valid.
5. Equation C: [tex]\(42 \div b = 8\)[/tex]
This implies dividing the area by the base equates the height, but doesn't match the operations needed for the area of a triangle in this problem. Hence, it's not accurate.
6. Equation F: [tex]\(b \div 8 = 42\)[/tex]
This equation suggests that dividing the base by the height equals the area, which doesn't correlate with the area formula for a triangle. Therefore, it is incorrect.
The correct equations for the given triangle's dimensions are:
- A. [tex]\(42 = \frac{1}{2} \cdot b \cdot 8\)[/tex]
- D. [tex]\(4 \cdot b = 42\)[/tex]
