\( \triangle C A B \) is right-angled at \( A \), \( E \in \overline{\mathrm{BC}} \) such that \( \overline{\mathrm{AE}} \perp \overline{\mathrm{BC}} \), D is the midpoint of \( \overline{\mathrm{BC}}, \mathrm{AE}=9.6 \mathrm{~cm} \). and \( \mathrm{BC}=20 \mathrm{~cm} \). Find : The length of each of \( \overline{\mathrm{AB}} \) and \( \overline{\mathrm{AC}} \)

We start with a right triangle \(\triangle CAB\) with the right angle at \(A\). Let the legs be \(\overline{AB}\) and \(\overline{AC}\) and the hypotenuse be \(\overline{BC}\) with \[ BC = 20\text{ cm}. \] An altitude from \(A\) to \(\overline{BC}\) meets \(BC\) at \(E\) and has length \[ AE = 9.6\text{ cm}. \] In any right triangle, the altitude to the hypotenuse divides the hypotenuse into two segments (say, \(BE\) and \(EC\)) which satisfy \[ AE^2 = BE \times EC. \] Let \[ BE = x \quad \text{and} \quad EC = 20-x. \] Then \[ 9.6^2 = x(20-x). \] Calculating, \[ 92.16 = x(20-x). \] This gives the quadratic equation \[ x^2 -20x + 92.16 = 0. \] The discriminant of this quadratic is \[ \Delta = 20^2 - 4(1)(92.16) = 400 - 368.64 = 31.36, \] so \[ \sqrt{\Delta} = 5.6. \] Thus the solutions are \[ x = \frac{20 \pm 5.6}{2}. \] This leads to: \[ x = \frac{25.6}{2} = 12.8 \quad \text{or} \quad x = \frac{14.4}{2} = 7.2. \] It does not matter which segment is labeled \(BE\) or \(EC\) because the legs will simply swap. The key extra property in a right triangle is that each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg. That is, we have: \[ AB^2 = BC \times BE \quad \text{and} \quad AC^2 = BC \times EC. \] Taking the case \(BE = 7.2\) and \(EC = 12.8\), we obtain: \[ AB^2 = 20 \times 7.2 = 144 \quad \Longrightarrow \quad AB = \sqrt{144} = 12\text{ cm}, \] \[ AC^2 = 20 \times 12.8 = 256 \quad \Longrightarrow \quad AC = \sqrt{256} = 16\text{ cm}. \] Thus, the lengths of the legs are: \[ \overline{AB} = 12\text{ cm} \quad \text{and} \quad \overline{AC} = 16\text{ cm}. \] Alternatively, if the segments are interchanged (i.e. \(BE=12.8\) and \(EC=7.2\)), then the leg lengths would be swapped, but the triangle would still have legs of 12 cm and 16 cm.