What is the height drawn to the diagonal of the square from one of its vertices if the length of the diagonal is 6 cm?

All sides of a square are equal. Hence, the diagonal divides the square into two isosceles right triangles. Let the side of the square be x cm, then by the Pythagorean theorem we compose the equation and solve it:

x² + x² = 6²,

2x² = 36,

x² = 36/2,

x² = 18,

x = √18,

x = 3√2 cm.

The area of ​​this particular triangle is:

S = ½ * x².

Let’s find it:

S = ½ * (3√2) ² = ½ * 9 * 2 = 9 cm².

On the other hand, the area of ​​the same triangle is:

S = ½ah, where a is the hypotenuse (and the diagonal of the square) of the triangle, h is the height drawn to the hypotenuse.

Hence the height is equal to:

h = 2S / a.

Find the height:

h = 2 * 9/6 = 3 cm.

This means that option a is correct.

Answer: the height drawn to the diagonal of the square is 3 cm.