Certainly, let's analyze the problem step-by-step to determine if Melinda's statements are correct.
Given:
1. The two sides of a right triangle are both given: [tex]\( a = 6 \)[/tex] feet and [tex]\( b = 8 \)[/tex] feet.
2. Melinda mentioned the area of the garden is 20 square feet.
We know the area of a triangle can be calculated using the formula:
[tex]\[ \text{Area} = \frac{1}{2} \times a \times b \][/tex]
Using the given side lengths [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ \text{Area} = \frac{1}{2} \times 6 \times 8 \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \times 48 \][/tex]
[tex]\[ \text{Area} = 24 \, \text{square feet} \][/tex]
So, Melinda's statement about the area being 20 square feet appears to be incorrect since the calculated area is 24 square feet.
Next, we need to calculate the length of the third side, which we'll denote as [tex]\( c \)[/tex]. Since this is a right triangle, we can use the Pythagorean theorem:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Substituting the given values:
[tex]\[ 6^2 + 8^2 = c^2 \][/tex]
[tex]\[ 36 + 64 = c^2 \][/tex]
[tex]\[ 100 = c^2 \][/tex]
Taking the square root of both sides:
[tex]\[ c = \sqrt{100} \][/tex]
[tex]\[ c = 10 \, \text{feet} \][/tex]
Now let’s evaluate each statement:
1. "Melinda is correct since the area is 20 square feet, which is" — We found the area to be 24 square feet, not 20 square feet. Thus, this statement is incorrect.
2. "Melinda is incorrect since the length of the third side is 13 f" — We found the length of the third side is 10 feet, not 13 feet. Thus, this statement is incorrect.
3. "Melinda is incorrect since the length of the third side is 39" — Again, the length of the third side is 10 feet, not 39 feet. Thus, this statement is incorrect.
4. "Melinda is correct since the length of the third side is 89 fe" — Finally, the length of the third side is 10 feet, not 89 feet. Thus, this statement is incorrect.
So, all given statements are incorrect based on the computed values. To summarize, the true statements regarding the garden are:
- The area of the garden is 24 square feet.
- The length of the third side of the triangle is 10 feet.
Melinda's assumptions about the area and the lengths given in the options do not match the correct calculations.
