use the triangle area formula $a = \frac{1}{2}bh$ to fill in the chart below.
label the base and height in the triangles to the right.
16. pamela wants to plant a triangular garden in her backyard and has 45 square feet of soil to use. she wants the base of her garden to line up against the back of her shed which is 10ft long. what will be the height of her garden?
17. if pamela increases the area of her garden but keeps the length of the base then same (i.e.; she still wants the base of the garden to line up against the shed) what will she have to do to the height of the garden?
18. howard is laying triangular tiles in his bathroom. the area of each tiles is 6 square inches and the height is 4 inches. what is the length of the base of each tile?
19. if howard swaps out his tiles for ones that have a height bigger than 4 inches but wants to keep the area the same what will he have to do to the base of each tile?
20. sam geometer is drawing a model for his father’s tree farm. his model trees must all fit on a piece of paper that is 3 ft. x 3 ft. he started making a chart to help him keep track of all the trees, but forgot to fill in everything as he went. he needs to finish the chart to show his father, but only has a short time to do it. he uses the area formula for a triangle and manipulates it so he can solve for each column by just putting the numbers in the calculator to find the missing piece of information.
a) what are the equations sam used for each column? area column $a = \frac{1}{2}bh$ base column $b = $ height column $h = $
b) finish filling in sam’s chart.
| $a = \text{area}$ | $b = \text{base}$ | $h = \text{height}$ |
| --- | --- | --- |
| | 10 cm | 5 cm |
| | 2 cm | 8 cm |
| | 8 cm | 19 cm |
| $25\ \text{cm}^2$ | 10 cm | |
| $10\ \text{cm}^2$ | 2 cm | |
| $42\ \text{cm}^2$ | 6 cm | |
| $20\ \text{cm}^2$ | | 8 cm |
| $90\ \text{cm}^2$ | | 12 cm |
| $110.5\ \text{cm}^2$ | | 13 cm |

Question

use the triangle area formula $a = \frac{1}{2}bh$ to fill in the chart below.
label the base and height in the triangles to the right.
16. pamela wants to plant a triangular garden in her backyard and has 45 square feet of soil to use. she wants the base of her garden to line up against the back of her shed which is 10ft long. what will be the height of her garden?
17. if pamela increases the area of her garden but keeps the length of the base then same (i.e.; she still wants the base of the garden to line up against the shed) what will she have to do to the height of the garden?
18. howard is laying triangular tiles in his bathroom. the area of each tiles is 6 square inches and the height is 4 inches. what is the length of the base of each tile?
19. if howard swaps out his tiles for ones that have a height bigger than 4 inches but wants to keep the area the same what will he have to do to the base of each tile?
20. sam geometer is drawing a model for his father’s tree farm. his model trees must all fit on a piece of paper that is 3 ft. x 3 ft. he started making a chart to help him keep track of all the trees, but forgot to fill in everything as he went. he needs to finish the chart to show his father, but only has a short time to do it. he uses the area formula for a triangle and manipulates it so he can solve for each column by just putting the numbers in the calculator to find the missing piece of information.
a) what are the equations sam used for each column? area column $a = \frac{1}{2}bh$ base column $b = $ height column $h = $
b) finish filling in sam’s chart.
| $a = \text{area}$ | $b = \text{base}$ | $h = \text{height}$ |
| --- | --- | --- |
|  | 10 cm | 5 cm |
|  | 2 cm | 8 cm |
|  | 8 cm | 19 cm |
| $25\ \text{cm}^2$ | 10 cm |  |
| $10\ \text{cm}^2$ | 2 cm |  |
| $42\ \text{cm}^2$ | 6 cm |  |
| $20\ \text{cm}^2$ |  | 8 cm |
| $90\ \text{cm}^2$ |  | 12 cm |
| $110.5\ \text{cm}^2$ |  | 13 cm |

Accepted Answer

### Question 16
# Explanation:
## Step1: Identify known values
We know the area formula for a triangle is $ A=\frac{1}{2}bh $. Pamela has an area $ A = 45 $ square feet and a base $ b=10 $ ft. We need to find the height $ h $.
## Step2: Rearrange the formula to solve for $ h $
Starting with $ A=\frac{1}{2}bh $, multiply both sides by 2: $ 2A = bh $. Then divide both sides by $ b $: $ h=\frac{2A}{b} $.
## Step3: Substitute the values
Substitute $ A = 45 $ and $ b = 10 $ into the formula: $ h=\frac{2\times45}{10} $.
## Step4: Calculate the result
$ 2\times45 = 90 $, then $ \frac{90}{10}=9 $.
# Answer:
The height of her garden will be 9 feet.

### Question 17
# Explanation:
## Step1: Analyze the area formula
The area formula for a triangle is $ A=\frac{1}{2}bh $. Here, the base $ b $ is kept constant.
## Step2: Relate area and height
Since $ A=\frac{1}{2}bh $ and $ b $ is constant, we can see that $ A $ is directly proportional to $ h $ (because $ A = (\frac{1}{2}b)h $, and $ \frac{1}{2}b $ is a constant factor).
## Step3: Determine the change in height
If the area $ A $ increases while the base $ b $ remains the same, and $ A $ is directly proportional to $ h $, then to increase $ A $, the height $ h $ must also increase.
# Answer:
She will have to increase the height of the garden.

### Question 18
# Explanation:
## Step1: Identify known values
We know the area $ A = 6 $ square inches and the height $ h = 4 $ inches. We use the formula $ A=\frac{1}{2}bh $ to find the base $ b $.
## Step2: Rearrange the formula to solve for $ b $
Starting with $ A=\frac{1}{2}bh $, multiply both sides by 2: $ 2A = bh $. Then divide both sides by $ h $: $ b=\frac{2A}{h} $.
## Step3: Substitute the values
Substitute $ A = 6 $ and $ h = 4 $ into the formula: $ b=\frac{2\times6}{4} $.
## Step4: Calculate the result
$ 2\times6 = 12 $, then $ \frac{12}{4}=3 $.
# Answer:
The length of the base of each tile is 3 inches.

### Question 19
# Explanation:
## Step1: Analyze the area formula
The area formula for a triangle is $ A=\frac{1}{2}bh $. Here, the area $ A $ is kept constant.
## Step2: Relate base and height
We can rewrite the formula as $ A=\frac{1}{2}bh\implies bh = 2A $ (since $ 2A $ is a constant when $ A $ is constant). So, $ b $ and $ h $ are inversely proportional (because their product is constant).
## Step3: Determine the change in base
If the height $ h $ increases while the area $ A $ remains the same, and $ b $ and $ h $ are inversely proportional, then to keep the product $ bh $ constant, the base $ b $ must decrease.
# Answer:
He will have to decrease the length of the base of each tile.

### Question 20a
# Explanation:
## Step1: Start with the area formula
The area formula for a triangle is $ A=\frac{1}{2}bh $.
## Step2: Solve for the base $ b $
To solve for $ b $, we start with $ A=\frac{1}{2}bh $. Multiply both sides by 2: $ 2A = bh $. Then divide both sides by $ h $: $ b=\frac{2A}{h} $.
## Step3: Solve for the height $ h $
To solve for $ h $, we start with $ A=\frac{1}{2}bh $. Multiply both sides by 2: $ 2A = bh $. Then divide both sides by $ b $: $ h=\frac{2A}{b} $.
# Answer:
- For the Base Column: $ b = \frac{2A}{h} $
- For the Height Column: $ h=\frac{2A}{b} $

### Question 20b
We will use the formulas $ A=\frac{1}{2}bh $, $ b = \frac{2A}{h} $, and $ h=\frac{2A}{b} $ to fill in the chart.

1. **Row 1: $ b = 10 $ cm, $ h = 5 $ cm**
   - $ A=\frac{1}{2}\times10\times5 = 25 $ $ \text{cm}^2 $
2. **Row 2: $ b = 2 $ cm, $ h = 8 $ cm**
   - $ A=\frac{1}{2}\times2\times8 = 8 $ $ \text{cm}^2 $
3. **Row 3: $ b = 8 $ cm, $ h = 19 $ cm**
   - $ A=\frac{1}{2}\times8\times19 = 76 $ $ \text{cm}^2 $
4. **Row 4: $ A = 25 $ $ \text{cm}^2 $, $ b = 10 $ cm**
   - $ h=\frac{2\times25}{10}=5 $ cm
5. **Row 5: $ A = 10 $ $ \text{cm}^2 $, $ b = 2 $ cm**
   - $ h=\frac{2\times10}{2}=10 $ cm
6. **Row 6: $ A = 42 $ $ \text{cm}^2 $, $ b = 6 $ cm**
   - $ h=\frac{2\times42}{6}=14 $ cm
7. **Row 7: $ A = 20 $ $ \text{cm}^2 $, $ h = 8 $ cm**
   - $ b=\frac{2\times20}{8}=5 $ cm
8. **Row 8: $ A = 90 $ $ \text{cm}^2 $, $ h = 12 $ cm**
   - $ b=\frac{2\times90}{12}=15 $ cm
9. **Row 9: $ A = 110.5 $ $ \text{cm}^2 $, $ h = 13 $ cm**
   - $ b=\frac{2\times110.5}{13}=\frac{221}{13}=17 $ cm

The filled - in chart is:

| $ A=\text{area} $ | $ b = \text{base} $ | $ h=\text{height} $ |
|----|----|----|
| $ 25 \text{cm}^2 $ | $ 10 \text{cm} $ | $ 5 \text{cm} $ |
| $ 8 \text{cm}^2 $ | $ 2 \text{cm} $ | $ 8 \text{cm} $ |
| $ 76 \text{cm}^2 $ | $ 8 \text{cm} $ | $ 19 \text{cm} $ |
| $ 25 \text{cm}^2 $ | $ 10 \text{cm} $ | $ 5 \text{cm} $ |
| $ 10 \text{cm}^2 $ | $ 2 \text{cm} $ | $ 10 \text{cm} $ |
| $ 42 \text{cm}^2 $ | $ 6 \text{cm} $ | $ 14 \text{cm} $ |
| $ 20 \text{cm}^2 $ | $ 5 \text{cm} $ | $ 8 \text{cm} $ |
| $ 90 \text{cm}^2 $ | $ 15 \text{cm} $ | $ 12 \text{cm} $ |
| $ 110.5 \text{cm}^2 $ | $ 17 \text{cm} $ | $ 13 \text{cm} $ |

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QUESTION IMAGE

Question

Explanation:

Question 16

Step1: Identify known values

Step2: Rearrange the formula to solve for \( h \)

Step3: Substitute the values

Step4: Calculate the result

Step1: Analyze the area formula

Step2: Relate area and height

Step3: Determine the change in height

Step1: Identify known values

Step2: Rearrange the formula to solve for \( b \)

Step3: Substitute the values

Step4: Calculate the result

Answer:

Question 17

$a = \text{area}$	$b = \text{base}$	$h = \text{height}$
	2 cm	8 cm
	8 cm	19 cm
$25\\ \text{cm}^2$	10 cm
$10\\ \text{cm}^2$	2 cm
$42\\ \text{cm}^2$	6 cm
$20\\ \text{cm}^2$		8 cm
$90\\ \text{cm}^2$		12 cm
$110.5\\ \text{cm}^2$		13 cm