The segment MH intersects the plane at point K. From the ends of this segment,

The segment MH intersects the plane at point K. From the ends of this segment, the prependiculars MM1 and HH1 are lowered onto the plane. Find the length of M1H1 if MM1 = 4cm, MK = 5cm, HH1 = 12cm

ΔММ1К is similar to ΔНН1К in two angles (If two corners of one triangle are similar to two corners of another triangle, then such triangles are similar). The angles MKM1 and NKN1 are equal as vertical. Angles M1 and H1 are straight.

In such triangles, the sides are proportional.

MM1 / HH1 = KM1 / KH1;

Let’s find КМ1 from rectangular ΔММ1К according to the Pythagorean theorem.

KM1 ^ 2 = KM ^ 2 – MM1 ^ 2;

KM1 ^ 2 = 5 ^ 2 – 4 ^ 2;

KM1 ^ 2 = 9;

KM = 3 (cm).

Substitute the known ones in ММ1 / НН1 = КМ1 / КН1;

4/12 = 3 / KH1;

4 * KH1 = 12 * 3;

4 * KH1 = 36;

KH1 = 36: 4 = 9 (cm).

M1H1 = KM1 + KH1;

M1H1 = 3 + 9 = 12 (cm).

Answer. 12 cm.