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## Question 8
To determine the number of fish in the tank, we need to use the given volume of the fish tank and the density of fish per cubic foot. Here's the step-by-step process:
1. Volume of the Fish Tank: We know that the volume of the tank is 50 cubic feet.
2. Density of Fish: The density of fish is given as [tex]\(0.2 \frac{\text{fish}}{\text{cubic feet}}\)[/tex].
Now, we need to calculate the total number of fish using the formula:
[tex]\[ \text{Number of Fish} = \text{Volume of Tank} \times \text{Density of Fish} \][/tex]
Substituting in the given values:
[tex]\[ \text{Number of Fish} = 50 \, \text{cubic feet} \times 0.2 \frac{\text{fish}}{\text{cubic feet}} \][/tex]
Multiplying these values together:
[tex]\[ \text{Number of Fish} = 50 \times 0.2 = 10 \][/tex]
Thus, the number of fish in the tank is 10.
Answer: 10
## Question 9
To prove that quadrilateral [tex]\(ABCD\)[/tex] is a parallelogram using coordinate geometry, we can use several geometric properties of parallelograms. Here’s the detailed reasoning:
1. Opposite Sides Parallel and Equal:
- Show that opposite sides are both parallel and equal in length.
- Calculate the slopes of [tex]\(AB\)[/tex], [tex]\(BC\)[/tex], [tex]\(CD\)[/tex], and [tex]\(DA\)[/tex]. If the slopes of [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex] are equal and the slopes of [tex]\(BC\)[/tex] and [tex]\(DA\)[/tex] are equal, then opposite sides are parallel.
- Then compute the lengths of [tex]\(AB\)[/tex], [tex]\(BC\)[/tex], [tex]\(CD\)[/tex], and [tex]\(DA\)[/tex]. If [tex]\(AB = CD\)[/tex] and [tex]\(BC = DA\)[/tex], then opposite sides are equal.
2. Diagonals Bisect Each Other:
- Show that the diagonals [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] bisect each other.
- Calculate the midpoints of diagonals [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex]. If the midpoints are the same, then the diagonals bisect each other.
3. Consecutive Angles Supplementary (optional):
- Compute the angles between consecutive sides. If consecutive angles sum up to 180 degrees, then the angles are supplementary.
Using one or a combination of these properties, you can prove that the quadrilateral [tex]\(ABCD\)[/tex] is a parallelogram. Usually, the most straightforward approach is to show the opposite sides are parallel and equal, or to show the diagonals bisect each other.
Answer: Which statement explains how you could use coordinate geometry to prove that quadrilateral [tex]\(ABCD\)[/tex] is a parallelogram? Using coordinate geometry, show either that the opposite sides of the quadrilateral are parallel and equal in length, or that the diagonals bisect each other.
To determine the number of fish in the tank, we need to use the given volume of the fish tank and the density of fish per cubic foot. Here's the step-by-step process:
1. Volume of the Fish Tank: We know that the volume of the tank is 50 cubic feet.
2. Density of Fish: The density of fish is given as [tex]\(0.2 \frac{\text{fish}}{\text{cubic feet}}\)[/tex].
Now, we need to calculate the total number of fish using the formula:
[tex]\[ \text{Number of Fish} = \text{Volume of Tank} \times \text{Density of Fish} \][/tex]
Substituting in the given values:
[tex]\[ \text{Number of Fish} = 50 \, \text{cubic feet} \times 0.2 \frac{\text{fish}}{\text{cubic feet}} \][/tex]
Multiplying these values together:
[tex]\[ \text{Number of Fish} = 50 \times 0.2 = 10 \][/tex]
Thus, the number of fish in the tank is 10.
Answer: 10
## Question 9
To prove that quadrilateral [tex]\(ABCD\)[/tex] is a parallelogram using coordinate geometry, we can use several geometric properties of parallelograms. Here’s the detailed reasoning:
1. Opposite Sides Parallel and Equal:
- Show that opposite sides are both parallel and equal in length.
- Calculate the slopes of [tex]\(AB\)[/tex], [tex]\(BC\)[/tex], [tex]\(CD\)[/tex], and [tex]\(DA\)[/tex]. If the slopes of [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex] are equal and the slopes of [tex]\(BC\)[/tex] and [tex]\(DA\)[/tex] are equal, then opposite sides are parallel.
- Then compute the lengths of [tex]\(AB\)[/tex], [tex]\(BC\)[/tex], [tex]\(CD\)[/tex], and [tex]\(DA\)[/tex]. If [tex]\(AB = CD\)[/tex] and [tex]\(BC = DA\)[/tex], then opposite sides are equal.
2. Diagonals Bisect Each Other:
- Show that the diagonals [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] bisect each other.
- Calculate the midpoints of diagonals [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex]. If the midpoints are the same, then the diagonals bisect each other.
3. Consecutive Angles Supplementary (optional):
- Compute the angles between consecutive sides. If consecutive angles sum up to 180 degrees, then the angles are supplementary.
Using one or a combination of these properties, you can prove that the quadrilateral [tex]\(ABCD\)[/tex] is a parallelogram. Usually, the most straightforward approach is to show the opposite sides are parallel and equal, or to show the diagonals bisect each other.
Answer: Which statement explains how you could use coordinate geometry to prove that quadrilateral [tex]\(ABCD\)[/tex] is a parallelogram? Using coordinate geometry, show either that the opposite sides of the quadrilateral are parallel and equal in length, or that the diagonals bisect each other.
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