We are given the following conditions about triangle $\triangle ABC$:

• $AB = 13$, $BC = 14$, and $AC = 15$.
• Point $D$ lies on $BC$ such that $AD \perp BC$, and $E$ is the midpoint of $AD$.
• Rectangle $CDEF$ is constructed such that points $C, D, E, F$ form its vertices.

We are tasked with finding the area of triangle $\triangle AEF$.

Step-by-Step Solution:

1. Right Triangle Identification: The triangle $ABC$ is a right triangle with $\angle B = 90^\circ$. This is because the sides 13, 14, and 15 follow the Pythagorean theorem, confirming it as a right triangle.

2. Coordinate Setup: Since $\triangle ABC$ is a right triangle, we can set up coordinates for the vertices:

   • $B(0, 0)$
   • $C(14, 0)$
   • $A(0, 13)$

   Now, $D$ lies on $BC$ and is the foot of the altitude from $A$. The altitude from $A$ to $BC$ implies that $AD \perp BC$.

3. Finding $D$: Since $BC$ is horizontal (on the x-axis), the line through $A$ perpendicular to $BC$ will have an undefined slope. Hence, the point $D$ must be directly beneath $A$ on the x-axis.

   • We can compute $D$'s coordinates by using similar triangles or by finding the length of the altitude using geometric properties of right triangles.

4. Midpoint $E$ and Rectangle: $E$ is the midpoint of $AD$, and rectangle $CDEF$ ensures that the lines $EF \parallel BC$ and $CF \parallel AD$, helping in setting up coordinates for all points.

5. Area of Triangle $\triangle AEF$: Once we calculate the coordinates of $A$, $E$, and $F$, the area of triangle $\triangle AEF$ can be computed using the formula for the area of a triangle with known vertices: $\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$ where $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ are the coordinates of points $A$, $E$, and $F$.

Let me go ahead and calculate the specific coordinates and the area for you!The $x$-coordinate of point $D$ is approximately $12.96$, and the length of the altitude $AD$ is approximately $5$ units (after rounding). Now, $E$ is the midpoint of $AD$, which we can calculate next.

The coordinates of $D$ are approximately $D(12.96, 0)$, and to find the coordinates of $E$, we use the midpoint formula:

$E = \left( \frac{x_A + x_D}{2}, \frac{y_A + y_D}{2} \right)$

Finally, we compute the area of $\triangle AEF$, using these coordinates.The coordinates of point $E$ (the midpoint of $AD$) are approximately $E(0.52, 6.5)$, and the coordinates of point $F$ are $F(14, 6.5)$, as it lies directly above $C$ at the same height as $E$.

The area of triangle $\triangle AEF$ is approximately 43.81 square units.

Would you like more details on any part of the solution, or do you have any further questions?

Related Questions:

1. How would you compute the length of diagonal $AF$ in this setup?
2. What are the properties of a rectangle that help simplify such problems?
3. Can the area of rectangle $CDEF$ be easily computed from these coordinates?
4. How can you prove that $AD$ is perpendicular to $BC$ geometrically?
5. How do the coordinates of point $F$ change if the rectangle were not aligned with the axes?

Tip:

In problems involving geometry and coordinate planes, assigning specific coordinates to key points can simplify calculations, especially when dealing with perpendicular lines or shapes like rectangles.