you act as a BC math teacher of Grade 12, please help your students understand this question and find the solution for it step-by-step---**Question Stem:**
In the diagram, $\triangle BDF$ is equilateral and $\angle FAB = \angle BCD = \angle DEF = 90^\circ$. Also, $AB = 16$, $BC = 8$, $DE = 5$, and $FA = 13$. Determine the perimeter of hexagon $ABCDEF$.
**Diagram Description:**
* **Type:** Geometric diagram.
* **Main Elements:**
* A hexagon ABCDEF is shown with vertices labeled A, B, C, D, E, F in counterclockwise order.
* An equilateral triangle BDF is drawn inside the hexagon, connecting vertices B, D, and F.
* Right angle symbols are present at vertices A, C, and E, indicating that angles ∠FAB, ∠BCD, and ∠DEF are 90 degrees.
* Side lengths are labeled: AB = 16, BC = 8, DE = 5, FA = 13.
* Tick marks are shown on the sides BD, DF, and FB of the triangle, indicating that these sides are equal in length.
**Given Information:**
* Triangle BDF is equilateral.
* ∠FAB = 90°.
* ∠BCD = 90°.
* ∠DEF = 90°.
* AB = 16.
* BC = 8.
* DE = 5.
* FA = 13.
**Question Asked:**
Determine the perimeter of hexagon ABCDEF.
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Welcome to this geometry problem! We have a hexagon ABCDEF with some special properties. Triangle BDF inside the hexagon is equilateral, meaning all its sides are equal. We also have right angles at vertices A, C, and E. We're given four side lengths: AB equals 16, BC equals 8, DE equals 5, and FA equals 13. Our goal is to find the perimeter of the entire hexagon.
Now let's solve this step by step. First, we use the key property that triangle BDF is equilateral. This means all three sides BD, DF, and FB are equal. Let's call this common length s. Next, we focus on the right triangle FAB. Since angle FAB is 90 degrees, we can apply the Pythagorean theorem. We have FA equals 13 and AB equals 16. Using the formula FA squared plus AB squared equals FB squared, we get 13 squared plus 16 squared equals s squared. Calculating: 169 plus 256 equals 425. So s squared equals 425.
Now let's find the length of side CD using triangle BCD. Since angle BCD is 90 degrees, we can again apply the Pythagorean theorem. We know BC equals 8, and BD equals s where s squared equals 425 from our previous calculation. Using the formula BC squared plus CD squared equals BD squared, we get 8 squared plus CD squared equals 425. This gives us 64 plus CD squared equals 425. Solving for CD squared: CD squared equals 425 minus 64, which equals 361. Taking the square root, CD equals 19.
Now let's find the length of side EF using triangle DEF. Since angle DEF is 90 degrees, we apply the Pythagorean theorem once more. We know DE equals 5, and DF equals s where s squared equals 425. Using the formula DE squared plus EF squared equals DF squared, we get 5 squared plus EF squared equals 425. This gives us 25 plus EF squared equals 425. Solving for EF squared: EF squared equals 425 minus 25, which equals 400. Taking the square root, EF equals 20.
Now we can calculate the final answer! We have found all six side lengths of the hexagon. The given sides are: AB equals 16, BC equals 8, DE equals 5, and FA equals 13. The calculated sides are: CD equals 19 and EF equals 20. To find the perimeter, we add all six sides together: 16 plus 8 plus 19 plus 5 plus 20 plus 13 equals 81. Therefore, the perimeter of hexagon ABCDEF is 81 units.