Point O is the center of the square, MO is the perpendicular to its plane.

Point O is the center of the square, MO is the perpendicular to its plane. Find the distance from point M to the sides of a square if its area is 36 cm ^ 2, and MO = 4 cm.

Let ABCD be a square. The distance from point M to the sides of the square will be the perpendicular (MH) lowered to them. To find its length, let’s find out the distance (OH) from the center of the square to its sides. To do this, we first find the length of the side of the square.
The area of ​​a square is found by the formula:
S = a ^ 2,
where a is the side length of the square.
a ^ 2 = 36;
a = √36;
a = 6 cm.
The distance from the center of the square to its sides (OH) is the perpendicular to the side of the square, as well as the radius of the circle inscribed in the square:
r = a / 2;
r = 6/2;
r = 3 cm.
OH = 3 cm.
Consider a triangle MON: angle MON = 90 degrees (since MO is perpendicular to the plane of the square), MO = 4 cm and OH = 3 cm are legs, MH is the hypotenuse (as well as the distance from point M to the sides of the square). By the Pythagorean theorem:
MH = √ (MO ^ 2 + OH ^ 2);
МН = √ (4 ^ 2 + 3 ^ 2) = √ (16 + 9) = √25 = 5 (cm).
Answer: MH = 5 cm.