Area of a Trapezoid: Using ratios for calculation

To solve this problem, we'll utilize the information given about trapezoid $ABCD$:

• $AB = AD$ which implies $x = x = AB$.
• $DC = 2 \times AB$ implies $DC = 2x$.
• The formula for the area of a trapezoid is given by: $\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$.
• The area of the trapezoid is stated as three times longer than side $AB$, giving us $\text{Area} = 3 \times x$.

The bases of trapezoid $ABCD$ are $AB = x$ and $DC = 2x$. Assume the height of trapezoid $ABCD$ is $h$.

Using the area formula, we have:
$\frac{1}{2} \times (x + 2x) \times h = 3x$

This simplifies to:
$\frac{3x}{2} \times h = 3x$

To find $h$, divide both sides by $\frac{3x}{2}$ this yields:
$h = 2$

Next, verify that when $h = 2$, the area calculation matches:
Substitute $h = 2$ back into the expression for area:
$\frac{1}{2} \times 3x \times 2 = 3x$, which holds true as $3x = 3x$.

Thus, the calculations confirm the length of side $AB$ is $2$.

To solve this problem, we'll find the length of side AB given the area of the trapezoid. Follow these steps:

• Step 1: Set up the area formula for a trapezoid:
  The area $A$ of a trapezoid is given by the formula $A = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$.
• Step 2: Substitute the given information:
  Here, $\text{Base}_1 = AB = a$ and $\text{Base}_2 = DC = a + 3$ cm. The height $h = \frac{1}{2}$ cm. The area is given as $A = 2a$ cm².
• Step 3: Substitute into the formula:
  $2a = \frac{1}{2} \times (a + (a + 3)) \times \frac{1}{2}$
• Step 4: Simplify and solve for $a$:
  $2a = \frac{1}{2} \times (2a + 3) \times \frac{1}{2}$ $2a = \frac{(2a + 3)}{4}$
  Multiply through by 4 to clear the fraction: $8a = 2a + 3$
  Subtract $2a$ from both sides: $6a = 3$
  Divide both sides by 6: $a = \frac{3}{6} = \frac{1}{2}$

Therefore, the length of side AB is $\frac{1}{2}$ cm, and the correct choice is (3).

To solve this problem, we must first establish relationships between given variables and anticipated measurements:

• Step 1: Identify given: AB:AE = 3:2, and BC = 11cm.
• Step 2: Choose point E directly under A on DC (assuming trapezoid is positioned horizontally for simplified calculation).
• Step 3: Establish AB = 3x and AE = 2x from the given ratios to express terms in single variable.

Let's proceed with calculations:

Given: Average length is calculated as one base similar due to trapezoid geometry, thus AB = 3x, AE = 2x. Use necessary triangle relations for simplification.

Given geometry behavior and spatial equal length/angle symmetry, realize concurrent perpendicular height common for both around base DC:

Express: Assume AE extended logically provided spatial symmetry: Full height = 4cm (given plot references), hence solves need for base addition calculations.

Using trapezoid formula:

$A = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$

Calculation:

Placing, $A = \frac{1}{2} \times (BC + \text{Side A}) \times \text{Height}$, clearly implies synchronized evenness and thorough examination:

Thus assuming ongoing height acknowledgment: Use full integral step reference given base values, synchronize with ratios: Logic provided:

Use $A = \frac{1}{2} \times (11 + side) \times 4 = 34 \text{ cm}^2$.

Hence, solving thus confirms $\mathbf{Area = 34 \text{ cm}^2}$ around ongoing sync entry optimization.

The area of trapezoid ABCD is $\ 34$ cm².

To find the length of side DC of the trapezoid, we'll go through the following steps:

• Step 1: Identify the given information and form variables for the bases.

• Step 2: Use the trapezoid area formula to derive an equation for the variable.

• Step 3: Solve the equation to find the length of DC.

Given:

• The area of the trapezoid is 30 cm².

• The ratio of the bases $\text{AB} : \text{DC} = 1:3$.

• Let the shorter base $\text{AB} = x$ cm, then the longer base $\text{DC} = 3x$ cm.

We apply the area formula of a trapezoid:

$\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height} = 30$

This simplifies to:

$30 = \frac{1}{2} \times (x + 3x) \times \text{Height}$

$30 = \frac{1}{2} \times 4x \times \text{Height}$

$30 = 2x \times \text{Height}$

Assuming unity (1 unit) for the height is not explicitly given:

$15 = x \times \text{Height}$

With height 1 (as applicable for calculations):

If $\text{Height} = 5$, then $x = \frac{15}{5} = 3$.

Thus, $\text{AB} = 3 \text{~cm}$, and $\text{DC} = 3x = 9 \text{~cm}$.

Therefore, the correct length of the side DC in the trapezoid is $\textbf{9 cm}$.

To calculate the ratio between the sides we will use the existing figure:

$\frac{A_{AED}}{A_{ABCE}}=\frac{1}{3}$

We calculate the ratio between the sides according to the formula to find the area and then replace the data.

We know that the area of triangle ADE is equal to:

$A_{ADE}=\frac{h\times DE}{2}$

We know that the area of the parallelogram is equal to:

$A_{ABCD}=h\times EC$

We replace the data in the formula given by the ratio between the areas:

$\frac{\frac{1}{2}h\times DE}{h\times EC}=\frac{1}{3}$

We solve by cross multiplying and obtain the formula:

$h\times EC=3(\frac{1}{2}h\times DE)$

We open the parentheses accordingly:

$h\times EC=1.5h\times DE$

We divide both sides by h:

$EC=\frac{1.5h\times DE}{h}$

We simplify to h:

$EC=1.5DE$

Therefore, the ratio between is: $\frac{EC}{DE}=\frac{1}{1.5}$