This is a right triangle problem, and the goal is to solve the triangle $\triangle DEF$, meaning we need to find the missing side $e$ (the hypotenuse) and the unknown angles.

Given:

• $DE = 14.3$ (adjacent side to $\angle DFE$)
• $EF = 11.2$ (opposite side to $\angle DFE$)

Step 1: Solve for the hypotenuse $DF$ using the Pythagorean theorem:

$DF = \sqrt{DE^2 + EF^2}$

Substituting: $DF = \sqrt{(14.3)^2 + (11.2)^2}$ $DF = \sqrt{204.49 + 125.44} = \sqrt{329.93}$ $DF \approx 18.16$

So, $DF \approx 18.16$.

Step 2: Solve for the angles.

Finding $\angle DFE$: We use the tangent function since: $\tan(\angle DFE) = \frac{\text{opposite}}{\text{adjacent}} = \frac{EF}{DE}$ Substitute: $\tan(\angle DFE) = \frac{11.2}{14.3} \approx 0.7832$ Taking the arctangent: $\angle DFE = \tan^{-1}(0.7832) \approx 38.28^\circ$

Finding $\angle EDF$: The sum of angles in a triangle is $180^\circ$, and since one angle is $90^\circ$: $\angle EDF = 90^\circ - \angle DFE$ $\angle EDF = 90^\circ - 38.28^\circ \approx 51.72^\circ$

Final Solution:

1. $DF \approx 18.16$
2. $\angle DFE \approx 38.28^\circ$
3. $\angle EDF \approx 51.72^\circ$

Would you like further details or explanations?

Follow-up Questions:

1. How do you apply the Pythagorean theorem in other cases?
2. What trigonometric function works best when you know two sides?
3. How would you calculate angles if given all three sides?
4. Can you explain the relationship between the angles and sides of a triangle?
5. How does the tangent function help in solving right triangles?

Tip:

Always verify your calculations with a different method (like sine or cosine) to ensure accuracy!