Through the vertex K of the triangle MKP, a straight line KN is drawn perpendicular to the plane

| June 21, 2021 | education

Through the vertex K of the triangle MKP, a straight line KN is drawn perpendicular to the plane of the triangle. It is known that KN = 15cm, MK = KP = 10cm. MP = 12cm. Find the distance from point N to line MP

Let us draw the height from the vertex K of the triangle MKP to the base of the MP. Since the MKP triangle is isosceles, the KE height divides the MP base into two equal segments.

The segment NE is also perpendicular to MP and will be the shortest distance from point N to line MP.

ME = PE = MP / 2 = 12/2 = 6 cm.

Consider the formed triangle KEP, in which the angle KEP = 90, since KE is the height to MP. Then, by the Pythagorean theorem, KE ^ 2 = KP ^ 2 – EP ^ 2 = 10 ^ 2 – 6 ^ 2 = 100 – 36 = 64.

Consider a triangle EKN, in which the angle NKE = 90, since by condition, NK is perpendicular to the plane of the triangle MKP. Then, by the Pythagorean theorem, NE ^ 2 = NK ^ 2 + KE ^ 2 = 225 + 64 = 289.

NE = 17 cm.

Answer: NE = 17 cm.



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