# To the vertex C of the rectangle ABCD, the perpendicular CN is drawn to its plane, the end of N of which

To the vertex C of the rectangle ABCD, the perpendicular CN is drawn to its plane, the end of N of which is at a distance of 10cm, 15cm and 17cm from the other vertices. Find the length of the perpendicular CN.

We introduce the variable x and denote the height СN as such.
Consider three right-angled triangles: BCN, DCN, ACN.
In each of them, by condition, the hypotenuse and leg CN = x are known.
We find the second leg:
Triangle BCN:
BC² = BN² – CN² = 225 – x².
DCN triangle:
DC² = DN² – CN² = 100 – x².
Triangle ACN:
AC² = AN² – CN² = 289 – x².
Consider a right-angled triangle ADC and write down the Pythagorean theorem in it:
AC² = BC² + DC²
289 – x² = (225 – x²) + (100 – x²)
x² = 36
x = 6 (cm) – perpendicular СN.
Answer: the length of the perpendicular is 6 cm.

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