Area of a Deltoid: Using additional geometric shapes

Let's solve the problem step by step:

• Calculate the area of the rectangular garden:
  The garden has a length of 10 meters and a width of 6 meters. Thus, the area is given by: $\text{Area of rectangle} = 10 \, \text{m} \times 6 \, \text{m} = 60 \, \text{m}^2.$

• Consider the deltoid shape:
  The provided image suggests the deltoid is inscribed within the rectangle. If we assume the deltoid is two congruent triangles making up part of the rectangle, let's find the area of each triangle.

• Area of each triangle of the deltoid:
  Assume two symmetrical triangles split the rectangle, each covering 30 m² (half of the rectangle). Hence, the deltoid area is the total area: $\text{Area of deltoid} = \frac{60}{2} \, \text{m}^2 = 30 \, \text{m}^2.$

• Calculate the area of one tile:
  The tile dimensions are 3 meters by 2 meters, so the area is: $\text{Area of one tile} = 3 \, \text{m} \times 2 \, \text{m} = 6 \, \text{m}^2.$

• Determine the number of tiles needed:
  Divide the deltoid's area by the area of one tile: $\text{Number of tiles} = \frac{30 \, \text{m}^2}{6 \, \text{m}^2} = 5.$

Therefore, the number of tiles needed to complete the deltoid shape is 5.

We can calculate the area of rectangle ABCD as follows:

$20\times30=600$

Now let's divide the deltoid along its length and width and add the following points:

Finally, we can calculate the area of deltoid PMNK as follows:

$PMNK=\frac{PN\times MK}{2}=\frac{20\times30}{2}=\frac{600}{2}=300$

Given the problem, we are tasked to find the value of $x$ for a deltoid where the length is twice the width and the area is given. Let's proceed as follows:

• Step 1: In this deltoid problem, the diagonals correspond to length $2x$ and width $x$. The formula for the area of a deltoid in terms of its diagonals is $A = \frac{1}{2} \times d_1 \times d_2$.
• Step 2: Substitute the values. Thus, the area $16 = \frac{1}{2} \times (2x) \times x$.
• Step 3: Simplify the equation: $16 = \frac{1}{2} \times 2x^2 = x^2$.
• Step 4: Solve for $x$: We find $x^2 = 16$, so $x = \sqrt{16}$.
• Step 5: Conclude $x = 4$.

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

To solve this problem, we'll use the properties of triangle and kite areas:

Firstly, we note that the area of triangle $ABD$ is given as $30 \, \text{cm}^2$.

We recognize that triangles $ABD$ and $ADC$ together form the kite with diagonal $AC$, and triangles $ADB$ and $BCD$ form diagonals where two triangles area will be half the kite’s complete area.

Now, find triangle $ABE$: $\frac{1}{2} \times AE \times BD = 30 \implies \frac{1}{2} \times 5 \times BD = 30 \implies BD = 12 \, \text{cm}$

Next, calculate diagonal $AC$ (sum of segments): $AE + EC = AC \implies 5 + 6 = AC \implies AC = 11 \, \text{cm}$

The area of kite $ABCD$ from diagonals $AC$ and $BD$: $\text{Area}_{kite} = \frac{1}{2} \times AC \times BD = \frac{1}{2} \times 11 \times 12 = 66 \, \text{cm}^2$

Therefore, the solution to the problem is 66 cm².

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

• Step 1: Calculate the side length of the square using the given area.
• Step 2: Determine the length of $BD$ using the side length.
• Step 3: Use the kite area formula with diagonals $BD$ and $AC=2x$.

Now, let's work through each step:

Step 1: The area of the square is 36 cm². The side length of the square, denoted as $s$, can be calculated by taking the square root of the area:

$s = \sqrt{36} = 6 \text{ cm}$

Step 2: To find diagonal $BD$, we use the relationship for the diagonal of a square in terms of its side:

$BD = s \sqrt{2}$. Given $s = 6$, we compute:

$BD = 6\sqrt{2} \text{ cm}$

Step 3: Now, we apply the formula for the area of a kite, which is $\frac{1}{2} \times d1 \times d2$, where $d1 = BD = 6\sqrt{2}$ and $d2 = AC = 2x$:

The area $A$ of the kite is:

$A = \frac{1}{2} \times 6\sqrt{2} \times 2x = 6\sqrt{2}x \text{ cm}^2$

Therefore, the area of the kite in terms of $x$ is $6\sqrt{2}x$ cm².