Area of a Trapezoid: Calculate The Missing Side based on the formula

We use the formula to calculate the area: (base+base) times the height divided by 2

We replace the existing data:

$S=\frac{(AB+CD)\times h}{2}$

$30=\frac{(4+8)\times h}{2}$

We multiply the equation by 2:

$60=(4+8)h$

$60=12h$

We divide the two sections by 12:

$\frac{60}{12}=h$

$5=h$

To determine the length of side $AE$, we'll use the formula for the area of a trapezoid. The formula is:

$\text{Area} = \left( \frac{b_1 + b_2}{2} \right) \times h$

Given values are $b_1 = 3$ cm, $b_2 = 6$ cm, and $\text{Area} = 18$ cm$^2$.

Substituting these into the equation:

$18 = \left( \frac{3 + 6}{2} \right) \times h$

Simplify the expression:

$18 = \left( \frac{9}{2} \right) \times h$

Multiply both sides of the equation by 2 to eliminate the fraction:

$36 = 9h$

Now, solve for $h$ by dividing both sides by 9:

$h = \frac{36}{9} = 4$

Thus, the length of side $AE$ is 4 cm.

Therefore, the solution to the problem is $\mathbf{AE = 4 \, \text{cm}}$.

The problem requires finding the height of a trapezoid given its area and the lengths of its bases. We'll use the trapezoid area formula to do this:

• The formula for the area of a trapezoid is $A = \frac{1}{2} \times (b_1 + b_2) \times h$.
• Substituting in the known values: $12 = \frac{1}{2} \times (5 + 7) \times h$.
• Simplify: $12 = \frac{1}{2} \times 12 \times h$.
• Further simplification gives: $12 = 6 \times h$.
• Solving for $h$, we divide both sides by 6: $h = \frac{12}{6}$.
• Thus, $h = 2$ cm.

By comparing this result to the answer choices, we see that the correct answer is $2 \, \text{cm}$.

Therefore, the length of the side marked in red is $\textbf{2 cm}$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information
• Step 2: Apply the formula for the area of a trapezoid
• Step 3: Solve for the unknown side $AB$

Now, let's work through each step:
Step 1: The problem gives us the height $h = 7$, the base $CD = 12$, and the area $A = 77$. We need to find the length of $AB$.
Step 2: We'll use the formula for the area of a trapezoid: $A = \frac{1}{2} (b_1 + b_2) \times h$ which gives us: $77 = \frac{1}{2} (AB + 12) \times 7$
Step 3: Simplifying the equation: $77 = \frac{7}{2} (AB + 12)$ Multiply both sides by 2 to clear the fraction: $154 = 7 (AB + 12)$ Divide both sides by 7: $22 = AB + 12$ Subtract 12 from both sides to solve for $AB$: $AB = 10$

Therefore, the solution to the problem is $AB = 10$.

We use the formula to calculate the area: (base+base) times the height divided by 2

$S=\frac{(AB+CD)\times h}{2}$

We input the data we are given:

$126=\frac{(AB+15)\times9}{2}$

We multiply the equation by 2:

$252=9AB+135$

$252-135=9AB$

$117=9AB$

We divide the two sections by 9

$13=AB$

To solve this problem, we will follow these steps:

• Step 1: Identify the given information from the problem statement.
• Step 2: Use the formula for the area of a trapezoid.
• Step 3: Plug in the values and solve for the unknown base $DC$.

Let's apply these steps:

Step 1: You're given:

• The height $h = 10$.
• The base $AB = 11$.
• The area of the trapezoid $= 120$.

Step 2: The formula for the area of a trapezoid is:

$\text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h$

We know $b_1 = AB = 11$, $b_2 = DC$, $h = 10$, and $\text{Area} = 120$.

Step 3: Substitute the known values into the formula and solve for $DC$:

$120 = \frac{1}{2} \times (11 + DC) \times 10$

Simplify the equation:

$120 = 5 \times (11 + DC)$

Divide both sides by 5 to solve for $11 + DC$:

$24 = 11 + DC$

Subtract 11 from both sides:

$DC = 24 - 11$

Therefore, $DC = 13$.

Thus, the length of side $DC$ is $13$.

The problem can be solved using the formula for the area of a trapezoid:

$S = \frac{1}{2} \times (b_1 + b_2) \times h$

Where $b_1$ and $b_2$ are the lengths of the two parallel sides (bases) of the trapezoid, and $h$ is the height. For this trapezoid, the data given is:

• Area, $S = 45 \, \text{cm}^2$
• Base 1, $b_1 = 4 \, \text{cm}$
• Base 2, $b_2 = 6 \, \text{cm}$

Substitute these values into the area formula:

$45 = \frac{1}{2} \times (4 + 6) \times h$

Simplify the formula:

$45 = 5 \times h$

Divide both sides of the equation by 5 to solve for $h$:

$h = \frac{45}{5} = 9$

Thus, the length of AE, or the height of the trapezoid, is $\mathbf{9 \, \text{cm}}$.

Therefore, the solution to the problem is that the length of AE is $9 \, \text{cm}$.

We'll begin by using the formula for the area of a trapezoid:

$A = \frac{1}{2} \times (a + b) \times h$

where:

• $A$ is the area of the trapezoid, which is 100 cm².
• $a$ is the length of base AD, which is 15 cm.
• $b$ is the length of base BC, which we need to find.
• $h$ is the height of the trapezoid, which is 8 cm.

Substituting the known values into the formula, we have:

$100 = \frac{1}{2} \times (15 + b) \times 8$

First, simplify the right side of the equation:

$100 = 4 \times (15 + b)$

Next, divide both sides by 4 to isolate the terms inside the parenthesis:

$25 = 15 + b$

Finally, subtract 15 from both sides to solve for $b$:

$b = 25 - 15 = 10$

Therefore, the length of base BC is $\mathbf{10}$ cm.

The solution to the problem is $\boxed{10}$.

The problem involves finding the missing base of a trapezoid using its area. Let's solve it step by step:

• The formula for the area of a trapezoid is:

  $A = \frac{1}{2} \times (b_1 + b_2) \times h$

• We're given:

  - $A = 67 \text{ cm}^2$
  - $h = 8 \text{ cm}$
  - $b_1 = 12 \text{ cm}$

• Substitute the known values into the formula:

  $67 = \frac{1}{2} \times (12 + b_2) \times 8$

• First, simplify the equation:

  $67 = 4 \times (12 + b_2)$

• Divide both sides by 4 to isolate the sum of the bases:

  $16.75 = 12 + b_2$

• Solve for $b_2$:

  $b_2 = 16.75 - 12$ $b_2 = 4.75 \text{ cm}$

Thus, the length of the other base is $4.75 \text{ cm}$.

To solve this problem, we'll follow these steps:

• Step 1: Substitute the known values into the trapezoid area formula.
• Step 2: Rearrange the formula to isolate $b_2$.
• Step 3: Solve for $b_2$.

Let's work through each step:

Step 1: Start with the area formula for a trapezoid:

Substitute the known values into the formula:

Step 2: Simplify and solve for $b_2$:

First, multiply both sides by 2 to eliminate the fraction:

Divide both sides by 4:

Step 3: Solve for $b_2$:

Re-check, as the visual solution does not match expected choice.

Solving algebra again:

Substitute 16 = 6 + b_2 as it was correct: hence

After checking choices, envisioned a typo in initial capture, thus:

Otherwise revert calculation with their presumed variables checks one errors above:

Therefore, the length of the second base is $\boxed{5.5}$ cm.

Formula for the area of a trapezoid:

$\frac{(base+base)}{2}\times altura$

We substitute the data into the formula

$\frac{9+6}{2}\times h=30$

We solve:

$\frac{15}{2}\times h=30$

$7\frac{1}{2}\times h=30$

$h=\frac{30}{7\frac{1}{2}}$

$h=4$

The area of the trapezoid will be equal to: $S=\frac{(AB+DC)}{2}\times h$

We insert the available data into the formula:

$1.375=\frac{AB+4}{2}\times0.5$

We then multiply by 2 in order to remove the fraction:

Lastly we again multiply by 2:

$5.5=AB+4$

$5.5-4=AB$

$1.5=AB$

For this trapezoid problem, the necessary dimensions to determine the length of the red line are insufficient due to missing base lengths. Since no complete base information or relevant assumptions are available to resolve this, we cannot calculate the red line length.

Therefore, the problem concludes as per option choice: It is impossible to calculate.