To solve the problem, let us analyze the given trapezoid $ABCD$ step by step:

Problem Setup:

1. Trapezoid:

   • $AB \parallel CD$, where $AB = a$ (larger base) and $CD = b$ (smaller base).
   • The segment in question is the line joining the midpoints of the bases $AB$ and $CD$, not the midline of the trapezoid.

2. Angles:

   • The sum of the angles adjacent to the larger base $AB$ is $90^\circ$. This means: $\angle DAB + \angle ABC = 90^\circ.$

3. Midpoints and Segment:

   • Let $E$ and $F$ be the midpoints of $CD$ and $AB$, respectively.
   • The task is to find the length of the segment $EF$, which joins these two midpoints.

Step 1: Coordinates of the Vertices

Let us place the trapezoid in a coordinate plane for simplicity:

• Place $A(0, 0)$ and $B(a, 0)$ along the larger base.
• Place $C(x, h)$ and $D(0, h)$ along the smaller base, where $h$ is the height of the trapezoid.

Step 2: Midpoints of the Bases

The midpoints $F$ and $E$ of the bases are:

• $F = \left(\frac{a}{2}, 0\right)$ (midpoint of $AB$).
• $E = \left(\frac{x + 0}{2}, h\right) = \left(\frac{x}{2}, h\right)$ (midpoint of $CD$).

Step 3: Length of $EF$

The length of segment $EF$ can be calculated using the distance formula: $EF = \sqrt{\left(\frac{a}{2} - \frac{x}{2}\right)^2 + \left(0 - h\right)^2}.$ Simplify the $x$-coordinate difference: $EF = \sqrt{\left(\frac{a - x}{2}\right)^2 + h^2}.$

Step 4: Relation Between $x$, $h$, and $a, b$

From the trapezoid geometry:

• The smaller base $CD = b = x$, so $x = b$.

Thus, the length of $EF$ becomes: $EF = \sqrt{\left(\frac{a - b}{2}\right)^2 + h^2}.$

Step 5: Determine the Height ($h$)

Using the condition that $\angle DAB + \angle ABC = 90^\circ$, we can relate the height $h$ and the bases:

• $\tan(\angle DAB) = \frac{h}{b}$, and $\tan(\angle ABC) = \frac{h}{a}$.
• Since $\tan(\angle DAB) \cdot \tan(\angle ABC) = 1$ (as their sum is $90^\circ$), we get: $\frac{h}{b} \cdot \frac{h}{a} = 1.$ Simplify: $\frac{h^2}{ab} = 1 \implies h^2 = ab.$

Step 6: Final Length of $EF$

Substitute $h^2 = ab$ into the formula for $EF$: $EF = \sqrt{\left(\frac{a - b}{2}\right)^2 + ab}.$

Final Answer:

The length of the segment joining the midpoints of the bases is: $EF = \sqrt{\left(\frac{a - b}{2}\right)^2 + ab}.$

Let me know if you'd like further clarifications or additional examples!

Related Questions:

1. How does the condition of the angles summing to $90^\circ$ affect the solution?
2. What happens if $a = b$ (the trapezoid becomes a parallelogram)?
3. How do we derive the height $h$ if the bases $a$ and $b$ and the angle $\angle DAB$ are given?
4. Can we compute the midline (average of bases) and compare it with $EF$?
5. How does the formula for $EF$ change if the trapezoid is not isosceles?

Tip:

Always use coordinate geometry for problems involving midpoints or distances in geometric shapes—it simplifies the calculations!