Area of a Deltoid: Using variables

To solve the question, we first need to remember the formula for the area of a kite:

Diagonal * Diagonal / 2

This means that if we substitute the given data we can see that:

a(2a+2)/2 = area of the kite

Let's remember that we are also given the area, so we'll put that in the equation too

a(2a+2)/2 = 6a

Now we have an equation that we can easily solve.

First, let's get rid of the fraction, so we'll multiply both sides of the equation by 2

a(2a+2)=6a*2
a(2a+2)=12a

Let's expand the parentheses on the left side of the equation

2a²+2a=12a

2a²=10a

Let's divide both sides of the equation by a

2a=10

Let's divide again by 2

a=5

And that's the solution!

To calculate $X$, follow these steps:

• Step 1: Use the area formula for a deltoid $\text{Area} = \frac{1}{2} \times d_1 \times d_2$.
• Step 2: Given the area $= 20 \, \text{cm}^2$, $d_1 = X$, and $d_2 = 5$, substitute these into the formula.
• Step 3: Set the equation: $20 = \frac{1}{2} \times X \times 5$.
• Step 4: Simplify and solve for $X$.

Now, let's solve:

Start with the equation $20 = \frac{1}{2} \times X \times 5$.

This simplifies to $20 = \frac{5X}{2}$.

Multiply both sides by 2 to eliminate the fraction:

$40 = 5X$.

Divide both sides by 5:

$X = \frac{40}{5}$.

Simplifying gives us $X = 8$.

Therefore, the length of diagonal $AC$ is $X = 8 \, \text{cm}$.

To solve this problem, we will utilize the formula for the area of a deltoid, which is $\frac{1}{2} \times AC \times DB$. Given:

• Diagonal $AC = 2X$
• Diagonal $DB = X$
• Area = $32 \, \text{cm}^2$

The formula for the area of the deltoid is:

$\text{Area} = \frac{1}{2} \times AC \times DB$

Substitute the given values into the formula:

$32 = \frac{1}{2} \times 2X \times X$

Simplify the equation:

$32 = \frac{1}{2} \times 2X^2$

$32 = X^2$

Solve for $X$ by taking the square root of both sides:

$X = \sqrt{32}$

Since $DB = X$, the length of diagonal $DB$ is $\sqrt{32}$.

Thus, the solution to the problem is $\sqrt{32}$.

To solve this problem, we'll use the formula relating the area of a deltoid to its diagonals:

• Step 1: Identify the given details:
• Diagonal AC = $X$.
• Diagonal DB = $3X$.
• Area = 27 cm$^2$.
• Step 2: Write down the formula for the area:

The area of a deltoid is given by:

$\text{Area} = \frac{1}{2} \times AC \times DB$

• Step 3: Substitute the values into the formula:

Substitute $AC = X$ and $DB = 3X$ into the equation:

$\frac{1}{2} \times X \times 3X = 27$

• Step 4: Simplify and solve for $X$:

First, simplify the left side:

$\frac{1}{2} \times 3X^2 = \frac{3}{2}X^2$

Thus, the equation becomes:

$\frac{3}{2}X^2 = 27$

Multiply both sides by 2 to clear the fraction:

$3X^2 = 54$

Divide both sides by 3:

$X^2 = 18$

Take the square root of both sides:

$X = \sqrt{18}$

Therefore, the length of diagonal AC is $\sqrt{18}$.

Let's solve this problem by working through the steps:

We are given a deltoid ABCD where:

• The length of diagonal BD is 5 cm.
• The sum of the segments forming diagonal AC is $a + 3a = 4a$.
• The area of the deltoid is 45 cm².

We use the area formula for a deltoid when the diagonals intersect at right angles:

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

Here, $d_1 = 5$ cm and $d_2 = 4a$. Substituting these values into the formula:

$\frac{1}{2} \times 5 \times 4a = 45$

Simplifying this equation:

$\frac{1}{2} \times 20a = 45$

$10a = 45$

Now, solve for $a$:

$a = \frac{45}{10}$

$a = 4.5$

Therefore, the length of segment $a$ is $4.5 \, \text{cm}$.

To solve this problem, we will calculate the value of $a$ using the given area of the deltoid and the known diagonal AC.

The formula for the area of a deltoid (kite) is:

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

Here, $d_1 = AC = 7$ cm, and $d_2 = BD = 9a$ cm. The area $S = 252$ cm².

Substitute the known values into the area formula:

$252 = \frac{1}{2} \times 7 \times 9a$

Simplify and solve for $a$:

$252 = \frac{63}{2} \times a$

$252 = 31.5a$

$a = \frac{252}{31.5}$

$a = 8$

Therefore, the value of the parameter $a$ is $8$.

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

• Step 1: Use the formula for the area of a deltoid.
• Step 2: Set up the equation using given values.
• Step 3: Solve for the variable $a$.

Let's go through each step:
Step 1: For a deltoid, the area $A$ can be calculated using:
$A = \frac{1}{2} \times AC \times BD$

Step 2: Substituting the known values into the expression gives us:
$182 = \frac{1}{2} \times 7a \times 13$

Step 3: Simplify and solve for $a$:
We first find $\frac{1}{2} \times 13 = 6.5$, so:
$182 = 6.5 \times 7a$
Solving for $a$, we have:
$182 = 45.5a$
$a = \frac{182}{45.5} = 4$

Therefore, the solution to the problem is $a = 4$.

To solve the problem, we'll follow the steps outlined:

• Step 1: Identify the given information.
• Step 2: Apply the appropriate formula for the area of the deltoid.
• Step 3: Solve for $b$ in terms of known values.

Let's break it down:

Step 1: We know the length of diagonal $BM = 4$ cm and the area of the deltoid is $144$ cm².

Step 2: The area of a deltoid is given by the formula:

$\text{Area} = \frac{1}{2} \times \text{Diagonal}_1 \times \text{Diagonal}_2$

Here, the diagonals correspond to line segments of the form $2b$ and $4b$ as represented in the setup of the problem.

Step 3: Substituting the values into the area formula, we have:

$\frac{1}{2} \times (2b) \times (4b) = 144$

Simplifying this, we get:

$4b^2 = 144 \quad \Rightarrow \quad b^2 = \frac{144}{4} = 36$

Therefore, solving for $b$, we find:

$b = \sqrt{36} = 6$

Thus, the value of $b$ is $6$.