From a point not lying in a certain plane, two segments 30 cm and 25 cm long are constructed

From a point not lying in a certain plane, two segments 30 cm and 25 cm long are constructed. The difference in the lengths of their projections is 11 cm. Find the distance from the point to the plane.

Let’s construct the perpendicular OM to the plane, which is the required distance.

Let the projection length OB = X cm, then OA = (11 – X) cm.

In the right-angled triangles OAM and OBM, by the Pythagorean theorem, we express the general leg OM.

OM ^ 2 = BM ^ 2 – OB ^ 2 = 625 – X ^ 2.

OM ^ 2 = AM ^ 2 – OA ^ 2 = 900 – (11 + X) ^ 2.

Then: 625 – X ^ 2 = 900 – (11 + X) ^ 2.

625 – X ^ 2 = 900 – 121 – 22 * X – X ^ 2.

22 * X = 154.

X = OB = 154/22 = 7 cm.

Then OM ^ 2 = 625 – 49 = 576.

OM = 24 cm.

Answer: From point M to plane 24 cm.