## 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.
