Point M is equidistant from the vertices of the rectangle, the lengths of the sides of which are 4 cm
Point M is equidistant from the vertices of the rectangle, the lengths of the sides of which are 4 cm and 19 cm. Find the distance from point M to the straight lines on which the sides of the rectangle lie if the distance from point M to the plane of the rectangle is 5 cm.
Let’s designate the rectangle as ABCD.
Let the sides AB = CD = 4 cm, and the sides BC = AD = 19 cm.
It is known that a point equidistant from the vertices of a rectangle lying in its plane corresponds to the point of intersection of the diagonals, which it divides in half. Let us denote it by point O. Since point M is also equidistant from the vertices of the rectangle, it can be argued that its projection onto the plane of the rectangle is point O, while MO is perpendicular to the plane of the rectangle. It is known from the properties of rectangles that the diagonals of the rectangle are equal. Also, the intersection point of the diagonals coincides with the intersection point of the middle lines of the rectangle. That is, if we lower the perpendicular from point O to each of the sides of the rectangle, this perpendicular will divide each side of the rectangle in half and will be equal to half of the side of the rectangle parallel to it. Let’s designate the points of intersection by perpendiculars from point O by points Ao, BO on the sides AB and BC. Wherein:
AAo = AoB = BOO = 4: 2 = 2 cm;
BBo = BoC = AoO = 19: 2 = 9.5 cm;
On the other two sides, the perpendiculars can be omitted – the numbers will be similar.
Consider a right-angled triangle AoMO. Side AoM will make the hypotenuse and the desired length from point M to the straight line on which side AB lies. In this triangle, both legs are known and you can use the Pythagorean theorem:
AoM = √‾ (OM ^ 2 + AoO ^ 2) = √‾ (5 ^ 2 + 9.5 ^ 2) = √‾ (25 + 90.25) = √‾115.25 ≈ 10.735455 …
The length is similar to the straight line on which the CD side lies.
The МВо triangle is also rectangular, the МВо segment makes up its hypotenuse and the required length from the point M to the straight line on which the BC side lies. Both legs are known, we find the hypotenuse:
BOM = √‾ (OM ^ 2 + BOO ^ 2) = √‾ (5 ^ 2 + 2 ^ 2) = √‾ (25 + 4) = √‾29 ≈ 5.385164 …
The length is similar to the straight line on which the AD side lies.
Thus, point M is equidistant from the pairwise parallel sides of the rectangle. From the sides AB and CD by about 10.74 cm, and from the sides BC and DA by about 5.39 cm.
