The sum of the hypotenuse CE and the leg CD of the right-angled triangle CDE is 31 cm

The sum of the hypotenuse CE and the leg CD of the right-angled triangle CDE is 31 cm, and from the difference is 3 cm. Find the distance from the vertex C to the line DE.

The distance from the vertex C to the straight line DE is equal to the length of the leg CD.
The sum of the lengths of the hypotenuse and leg is 31 cm, and their difference is 3 cm, we can draw up a system of equations:
1) CE-CD = 3;
2) CE + CD = 31.
In the first equation, we express the leg CD through the hypotenuse CE: CD = CE-3, and substitute the resulting expression into the second equation:
CE + CD = CE + CE-3 = 31;
2CE = 34;
CE = 17.
Hence CD = 17-3 = 14 cm.Consequently, the distance from the vertex C to the line DE is 14 cm.