The lengths of the sides of the rectangle are 8 and 6 cm. Through the point O of the intersection of its diagonals

The lengths of the sides of the rectangle are 8 and 6 cm. Through the point O of the intersection of its diagonals, a straight line OK is drawn, perpendicular to its plane. Find the distance from point K to the vertices of the triangle if OK = 12 cm.

Two adjacent sides of a rectangle and its diagonal form a right-angled triangle, in which the diagonal is the hypotenuse and the sides are the legs. Knowing that the lengths of the sides are 6 cm and 8 cm, by the Pythagorean theorem we can find the diagonal:

d² = 6² + 8² = 36 + 64 = 100 = 102;

d = 10 cm.

The diagonals of the rectangle are equal to each other and are halved at the intersection. Consider a right-angled triangle formed by a perpendicular OK, half a diagonal and an inclined one drawn from point K to the vertex of the rectangle. The square of the length of this oblique is defined as the sum of the squares of the lengths of the perpendicular and half of the diagonal:

l² = OK² + (d / 2) ² = 12² + 5² = 144 + 25 = 169 = 132;

l = 13 cm is the distance from point K to the vertices of the rectangle.