how to solve ABC is a right triangle with C as the right angle and sides AC=6 cm and BC=8 cm. Line segment CD is drawn perpendicular to both AC and BC at C. If CD=12 cm, find the distance from D to midpoint E of AB.

To find the distance from point D to the midpoint E of side AB, we can use the Pythagorean theorem and the concept of similar triangles.

First, let's draw the triangle ABC and the perpendicular line segment CD:

```
C
/ |
6 / | \
/ | \
A-------D------B
AC CD BC
```

Since CD is perpendicular to AC and BC, we can see that triangle ACD and triangle BCD are similar to triangle ABC.

From triangle ACD, we can find AD using the Pythagorean theorem:

AD^2 = AC^2 - CD^2
AD^2 = 6^2 - 12^2
AD^2 = 36 - 144
AD^2 = -108

Since the result is negative, we can conclude that triangle ACD does not exist, and the triangle ABC is not a right triangle. Therefore, we cannot determine the distance from D to the midpoint E of side AB.

To solve this problem, we need to apply the Pythagorean theorem and use the properties of a right triangle. Let's go step by step:

Step 1: Draw the triangle and label the given measurements:
- Triangle ABC with the right angle at C.
- Side AC measures 6 cm.
- Side BC measures 8 cm.
- Line segment CD is perpendicular to both AC and BC, and it measures 12 cm.

Step 2: Identify the right triangle and find the length of the third side AB:
Since triangle ABC is a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Applying the Pythagorean theorem, we have:
AB^2 = AC^2 + BC^2
AB^2 = 6^2 + 8^2
AB^2 = 36 + 64
AB^2 = 100

Taking the square root of both sides, we find:
AB = 10 cm

So, the length of side AB is 10 cm.

Step 3: Find the midpoint E of side AB:
To find the distance from point D to the midpoint E of side AB, we need to calculate half of the length of side AB.

Since AB measures 10 cm, half of this length is:
AB/2 = 10/2 = 5 cm

Therefore, the distance from point D to the midpoint E of side AB is 5 cm.

In summary, the distance from D to midpoint E of AB is 5 cm.

Our nice triangle has hypotenuse length = 10.

As we know, the median from C to AB is 1/2 AB, or 5.

So, we now have triangle AED with legs 5 and 12, so DE = 13.