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

To solve the problem, we need to remember the formula for the area of a rhombus:

The product of the diagonals multiplied together and then divided by 2.

Let's substitute in our data into the formula:

(8*X) = 60
2

Note that we can simplify the fraction, thus eliminating the denominator:

4X = 60

Let's finally divide the equation by 4 to get our answer:

X = 15

We substitute the data we have into the formula for the area of the kite:

$S=\frac{AC\times BD}{2}$

$42=\frac{AC\times14}{2}$

We multiply by 2 to remove the denominator:

 $14AC=84$

Then divide by 14:

$AC=6$

In a rhombus, the main diagonal crosses the second diagonal, therefore:

$AO=\frac{AC}{2}=\frac{6}{2}=3$

To solve for the length of side $AC$ in the deltoid $ABCD$, we will use the deltoid area formula:

The formula for the area of a deltoid is given by $\text{Area} = \frac{1}{2} \times d_1 \times d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals.

Given:

• The area of the deltoid: $\text{Area} = 20 \, \text{cm}^2$
• The length of diagonal $BD = 4 \, \text{cm}$.
• We need to find the length of diagonal $AC$.

Substitute the known values into the formula:

$20 = \frac{1}{2} \times AC \times 4$

Re-arrange the equation to solve for $AC$:

$20 = 2 \times AC$

Divide both sides by 2:

$AC = \frac{20}{2} = 10 \, \text{cm}$

Thus, the length of side $AC$ is $10 \, \text{cm}$.

The only choice matching this calculation is:

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

• Step 1: Utilize the formula for the area of a kite or deltoid, $S = \frac{1}{2} \cdot d_1 \cdot d_2$.
• Step 2: Substituting the known values into this formula.
• Step 3: Solve for the unknown diagonal $BD$.

Now, let's work through each step:
Step 1: The given side $AC$ acts as the first diagonal $d_1 = 10$ cm. The area $S = 40$ cm².
Step 2: Plug these values into the formula $S = \frac{1}{2} \cdot d_1 \cdot d_2$ which becomes $40 = \frac{1}{2} \cdot 10 \cdot BD$.
Step 3: Solving for $BD$ involves rearranging the equation: $40 = \frac{1}{2} \cdot 10 \cdot BD \implies 40 = 5 \cdot BD \implies BD = \frac{40}{5} = 8 \text{ cm}$

Therefore, the length of the side $BD$ is $8 \text{ cm}$.

To find the length of side $AC$, follow these steps:

• Step 1: Use the formula for the area of a deltoid: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$.
• Step 2: Set the known values into the equation, where $d_1 = BD = 6$ cm and the area is 54 cm$^2$.
• Step 3: Rearrange the formula to solve for $d_2$ (which is $AC$): $54 = \frac{1}{2} \times 6 \times AC$.
• Step 4: Simplify and solve for $AC$: $54 = 3 \times AC$.
• Step 5: Divide both sides by 3 to isolate $AC$: $AC = \frac{54}{3} = 18$ cm.

Therefore, the length of $AC$ is $18$ cm.

To solve this problem, we will compute the length of diagonal BD using the formula for the area of a deltoid:

• Step 1: Recall the formula for the area of a deltoid, which is given by:
$\text{Area} = \frac{1}{2} \times AC \times BD$
• Step 2: Substitute the known values into the formula:
$72 = \frac{1}{2} \times 9 \times BD$
• Step 3: Solve the equation for BD:

First, multiply both sides of the equation by 2 to clear the fraction:

$144 = 9 \times BD$

Next, divide both sides by 9 to isolate BD:

$BD = \frac{144}{9}$
• Step 4: Perform the division:
$BD = 16$

Thus, the length of diagonal BD is $16$ cm.

This conclusion matches the possible answer choice 4:

The correct choice is (4): $16$ cm.

To solve this problem, we'll use the formula for the area of a deltoid:

• $\text{Area} = \frac{1}{2} \times d_1 \times d_2$

Let's work through the steps:

Step 1: Write down the formula for the area of the deltoid. The area $S$ is given as:

$44 = \frac{1}{2} \times 11 \times BD$

Step 2: Rearrange this equation to solve for the unknown diagonal $BD$:

$44 \times 2 = 11 \times BD$

$88 = 11 \times BD$

Step 3: Divide both sides by 11 to find the length of $BD$:

$BD = \frac{88}{11} = 8$ cm

Therefore, the solution to the problem is $BD = 8$ cm.

To solve for the length of side AC in the deltoid:

• Step 1: Identify the formula for the area, which is given by:
  $\text{Area} = \frac{1}{2} \times BD \times AC$ where $BD$ and $AC$ are the diagonals.
• Step 2: Substitute known values into the formula:
  Given that $BD = 7\, \text{cm}$ and $\text{Area} = 49\, \text{cm}^2$, we have:
  $49 = \frac{1}{2} \times 7 \times AC$
• Step 3: Solve the equation for $AC$:
  Multiply both sides of the equation by 2 to eliminate the fraction:
  $98 = 7 \times AC$ Divide both sides by 7:
  $AC = \frac{98}{7} = 14$

Therefore, the length of the side $AC$ is $14 \, \text{cm}$.

To find the length of diagonal $BD$, we will apply the formula for the area of a deltoid:

$\text{Area} = \frac{1}{2} \times \text{Diagonal 1} \times \text{Diagonal 2}$

In this problem, Diagonal 1 is $AC = 13$ cm, and Diagonal 2 is $BD$, which we are trying to find. The area is given as $39$ cm². Substituting these values into the formula, we get:

$39 = \frac{1}{2} \times 13 \times BD$

To solve for $BD$, multiply both sides by 2 to eliminate the fraction:

$78 = 13 \times BD$

Now, solve for $BD$ by dividing both sides by 13:

$BD = \frac{78}{13}$

Simplify to find:

$BD = 6$

Therefore, the length of diagonal $BD$ is $6$ cm.

To solve for $CO$, we will use the area formula for the deltoid:

• Step 1: Calculate full length of diagonal $BD$:

$BD = 2 \times BO = 2 \times 5 = 10 \text{ cm}$.

• Step 2: Use the kite area formula:

• $\text{Area} = \frac{1}{2} \cdot AC \cdot BD$.

Substitute known values into the formula:

$80 = \frac{1}{2} \cdot (6 + CO) \cdot 10$.

Step 3: Simplify and solve for $CO$:

$80 = 5 \cdot (6 + CO)$ leads to

$80 = 30 + 5CO$.

Solving for $CO$, we subtract 30 from both sides:

$50 = 5CO$,

$CO = \frac{50}{5} = 10$.

Therefore, the side $CO$ is 10 cm.

To solve for $BD$, the diagonal of the deltoid, follow these steps:

• Step 1: Recognize that the area of a deltoid with perpendicular diagonals is given by the formula $A = \frac{1}{2} \times d_1 \times d_2$.
• Step 2: Substitute the known values into the formula: $48 = \frac{1}{2} \times 6 \times BD$.
• Step 3: Simplify and solve for $BD$:

Substituting $AC = 6$ cm, we have:

$48 = \frac{1}{2} \times 6 \times BD$

Multiply both sides by 2 to clear the fraction:

$96 = 6 \times BD$

Divide both sides by 6 to solve for $BD$:

$BD = \frac{96}{6} = 16$ cm

Thus, the length of $BD$ is $16$ cm.

To find the length of side $AC$ in the given deltoid:

• Step 1: Write down the area formula for a kite: $S = \frac{1}{2} \times BD \times AC$, where $S$ is the area, and $BD$ and $AC$ are the diagonals.
• Step 2: Plug in the known values: $60 = \frac{1}{2} \times 12 \times AC$.
• Step 3: Simplify the equation: $60 = 6 \times AC$.
• Step 4: Solve for $AC$:

$AC = \frac{60}{6} = 10$ cm.

Therefore, the length of side $AC$ is $10$ cm.

To solve the problem of finding the length of the diagonal $BD$ in deltoid $ABCD$, where $AC = 8 \, \text{cm}$ and the area $S = 64 \, \text{cm}^2$, follow these steps:

• Step 1: Identify the given values: $AC = 8 \, \text{cm}$ and $S = 64 \, \text{cm}^2$.
• Step 2: Apply the area formula for a deltoid: $S = \frac{1}{2} \times AC \times BD$.
• Step 3: Substitute the known values into the formula.
• Step 4: Solve for $BD$.

Now, let's work through the calculation:

Given the formula for the area of a deltoid: $S = \frac{1}{2} \times AC \times BD$

Substitute the known values: $64 = \frac{1}{2} \times 8 \times BD$

To solve for $BD$, first multiply both sides by 2 to get rid of the fraction: $128 = 8 \times BD$

Now, divide both sides by 8 to isolate $BD$: $BD = \frac{128}{8} = 16$

Therefore, the length of $BD$ is $\textbf{16 cm}$.

To solve this problem, we'll proceed as follows:

• Step 1: Identify given information and formula to use: The area of a deltoid is given as $\text{Area} = \frac{1}{2} \times d_1 \times d_2$.
• Step 2: Plug in the known values: $27.5 = \frac{1}{2} \times 5.5 \times BD$.
• Step 3: Calculate the unknown length $BD$.

Now, let's work through each step:

Step 1: Given $\text{Area} = 27.5$ cm$^2$, $AC = 5.5$ cm, and the formula for the area of a deltoid: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ where $d_1 = AC$ and $d_2 = BD$.

Step 2: Use the formula with the given values:
$27.5 = \frac{1}{2} \times 5.5 \times BD$.

Step 3: Solve for $BD$:
Multiply both sides by 2 to eliminate the fraction:
$55 = 5.5 \times BD$.
Now, divide both sides by $5.5$:
$BD = \frac{55}{5.5}$.

Simplify $\frac{55}{5.5}$:
$BD = 10$ cm.

Therefore, the length of side $BD$ is $10$ cm.

To solve this problem, we'll apply the following steps:

• Step 1: Identify the given information: $BD = 15$ cm and the area is $60$ cm².

• Step 2: Use the formula for the area of a deltoid.

• Step 3: Solve for the unknown diagonal $AC$.

Now, let's work through each step:
Step 1: We know the area formula for a deltoid is given by: $\text{Area} = \frac{1}{2} \times AC \times BD$

Step 2: Substitute the given values into the formula: $60 = \frac{1}{2} \times AC \times 15$

Step 3: Simplify and solve for $AC$: $60 = \frac{15}{2} \times AC$
Multiply both sides by 2 to isolate $AC$: $120 = 15 \times AC$
Divide both sides by 15: $AC = \frac{120}{15} = 8$

Therefore, the length of the side $AC$ is $8$ cm.

To solve for the length of $AC$ in the deltoid:

• Given $MD = 3$ cm, which implies $BD = 2 \times MD = 6$ cm.
• The area of the deltoid is given by the formula: $\text{Area} = \frac{1}{2} \times AC \times BD$.

Putting the known values into the formula:
$72 = \frac{1}{2} \times AC \times 6$.

To isolate $AC$, multiply both sides by 2:

$144 = AC \times 6$.

Divide both sides by 6 to solve for $AC$:

$AC = \frac{144}{6} = 24 \, \text{cm}$.

Therefore, the length of the side $AC$ is $24 \, \text{cm}$.

To solve this problem, we'll employ the formula for the area of a kite or deltoid, which relates to its diagonals AC and BD.

The formula is:

Given that the diagonal BD consists of BM and MD, and BM = MD as M is the midpoint, we have:

Also, the area is given as 72 cm². We substitute into the area formula:

Simplifying the equation by multiplying through by 2 to eliminate the fraction:

Divide both sides by 4 to solve for AC:

Therefore:

Thus, the length of side AC is $\textbf{36 cm}$.

To calculate the length of the diagonal $AC$, we start by using the area formula for a deltoid, which involves its diagonals. The area $A$ of a deltoid is given by:

Given:

• The area of the deltoid $A = 28 \, \text{cm}^2$.
• The length of diagonal $DB = 4 \, \text{cm}$.

We can plug these values into the formula:

Solving for $AC$:

Divide both sides by $2$:

Therefore, the length of side $AC$ is 14 cm.

To solve for $AC$ in the deltoid, use the area formula:

• Step 1: The area of a deltoid is given by $A = \frac{1}{2} \times p \times q$, where $p$ and $q$ are diagonals.
• Step 2: For this problem, set $p = DB = 4$ cm, and $q = AC$ (unknown).
• Step 3: Substitute the values: $32 = \frac{1}{2} \times 4 \times AC$.

Now, solve for $AC$:
$32 = \frac{1}{2} \times 4 \times AC$
$32 = 2 \times AC$
Divide both sides by 2:
$AC = \frac{32}{2} = 16 \, \text{cm}$

The side $AC$ is therefore 16 cm.

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

• Step 1: Set up the equation using the area of the deltoid formula.
• Step 2: Substitute the known values and solve for the unknown diagonal $DB$.

Here's the step-by-step solution:

Step 1: The area of a deltoid can be calculated using the formula:
$\text{Area} = \frac{1}{2} \times d_1 \times d_2$
Given that $\text{Area} = 32$ cm² and $d_1 = AC = 8$ cm, we place these values into the equation:

Step 2: Substitute into the formula:
$32 = \frac{1}{2} \times 8 \times DB$

Step 3: Simplify the equation:
$32 = 4 \times DB$

Step 4: Solve for $DB$:
$DB = \frac{32}{4} = 8$

Therefore, the length of diagonal $DB$ is 8 cm.