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------------------------------------------------ The triangle has a height of 8 ft and an area of [tex]$42 \, \text{ft}^2$[/tex]. Select all the equations that represent the relationship between the dimensions of the triangle.

A. [tex]42 = \frac{1}{2} \cdot b \cdot 8[/tex]

B. [tex]42 \div w = \frac{1}{2} \cdot 8[/tex]

C. [tex]42 \div b = 8[/tex]

D. [tex]4 \cdot b = 42[/tex]

E. [tex]8 \cdot b = 42[/tex]

F. [tex]b \div 8 = 42[/tex]

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]