Perimeter of a Rectangle: Using congruence and similarity

Let's begin by observing triangle FCE and calculate side FC using the Pythagorean theorem:

$EC^2+FC^2=EF^2$

Let's begin by substituting all the known values into the formula:

$8^2+FC^2=10^2$

$64+FC^2=100$

$FC^2=100-64$

$FC^2=36$

Let's take the square root:

$FC=6$

Since we know that the triangles overlap:

$\frac{AD}{FC}=\frac{DE}{CE}=\frac{AE}{FE}$

Let's again substitute the known values into the formula:

$\frac{AD}{6}=\frac{16}{8}$

$AD=2\times6=12$

Finally let's calculate side CD:

$16+8=24$

Since in a rectangle each pair of opposite sides are equal, we can calculate the perimeter of rectangle ABCD as follows:

$12+24+12+24=24+48=72$

To find the perimeter of rectangle EDGF, we will follow these steps:

• Step 1: Recognize congruence $\Delta BCE \equiv \Delta FED$.
• Step 2: Use congruent triangles to find side lengths of ED and DF.
• Step 3: Calculate the perimeter using the formula for rectangles.

Now, let's proceed with the solution:

Step 1: Given $\Delta BCE$ and $\Delta FED$ are congruent triangles, their corresponding sides are equal. Thus, since $BC = 6$, it follows that $ED = BC = 6$.

Step 2: From the configuration of the rectangle, note that $FE$ (same as AE) is given to be 8, hence $GF = 8$ because EDGF is a rectangle.

Step 3: The perimeter of rectangle EDGF can be calculated using:

Therefore, the solution to the problem is that the perimeter of rectangle EDGF is 28.

To solve for the perimeter of rectangle $EFGD$, we follow these steps:

Step 1: Use the similarity of the triangles $\triangle BCD \sim \triangle FED$. This implies the sides are proportional:

Substitute the known lengths $CD = 8$ and $ED = 3$:

Solving for $EF$, cross-multiply:

Step 2: The length and width of rectangle $EFGD$ are $2.25$ (as calculated from $EF$) and $3$ (equal to $ED$).

Step 3: Calculate the perimeter $P$ of $EFGD$:

The perimeter of rectangle $EFGD$ is $10.5$.

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

• Step 1: Analyze the congruent triangles.

• Step 2: Use congruence to identify side lengths.

• Step 3: Calculate the perimeter of the rectangle.

Now, let's work through each step:
Step 1: The problem states that $\triangle AED \cong \triangle BCD$, which means corresponding sides and angles are equal.
Step 2: Identify corresponding sides from congruence:
- $AE = CD = 6$ because corresponding sides of congruent triangles are equal.
- $AD = BC$. Since $AB = 8$ and using the fact $\triangle AED$ and $\triangle BCD$ are congruent, $BD$ (the same as $AD$) must also equal 6 (from the congruence starting from point $D$).
- Thus, $AD = 6$.

Step 3: Calculate the perimeter: $AF = DE = 6$ from congruency and $AD = FE = 6$. Since both pairs of opposite sides of the rectangle are equal, the rectangle perimeter is:
$P = 2(AF + AD) = 2(6 + 8) = 2 \times 14 = 28$

Therefore, the perimeter of the rectangle $AFDE$ is $28$.