Aladder which is 3.4 meters long is leaning against a wall. its base is 1.6 meters from the wall. how many meters up the wall does the ladder reach? (no calculators on this one.)

Make an illustration of the problem as shown in the diagram of the attached picture. The ladder held up against the wall is the inclined line. Since the wall and the ground are perpendicular to each other, we can deduce that the right triangle formed by the wall, ladder and ground is a right triangle. So, we can use the pythagorean theorems.

For a right triangle, there are three sides with respect to a certain angle. The longest side of the triangle is called a hypotenuse, usually denoted as c. The other two legs are the opposite side and the adjacent side, respective to a certain angle. These two legs are denoted as sides a and b. The formula would be

a² + b² = c²
Substituting the values
a² + (1.6)² = (3.4)²
The unknown figure a will be replaced by the variable x, which is the height of the wall. Transposing the terms,

x = √(3.4² - 1.6²)

You can do this manually but it will take time. For your reference, the correct answer would be 

x = √9
x = 3

Therefore, the ladder reaches 3 meters up to the wall.