Question

КО – перпендикуляр к плоскости α, КМ и КР – наклонные, ОМ и ОР – их проекции на плоскость α, причём сумма их длин равна 15 см. Найти расстояние от точки К до плоскости α, если КМ = 15 см, КР = см.

Answer 1

Пусть МО=х, тогда ОР=15-х ОК^2=15^2-x^2 OK^2=300-(15-x)^2 Приравняем и решим уравнение х=5 Теперь найдем ОК^2=15^2-5^2=200 ОК=10*корень из 2

Answer 2

Task description

We have a problem described in Russian. The problem asks to find the distance from point K to plane α. The given information includes the lengths of KM and KR, the projections of OM and OR on the plane α, and the sum of their lengths.

Translation of the given information

Let's start by translating the given information into English for better understanding:

- KM = 15 cm (length of KM) - KR = ? cm (length of KR) - OM and OR are the projections of the vectors OM and OR on the plane α. - The sum of the lengths of OM and OR is 15 cm.

Solution

To solve this problem, we need to find the length of KR and then use it to calculate the distance from point K to plane α.

Let's assume that KR = x cm.

Since OM and OR are the projections of the vectors OM and OR on the plane α, we can use the concept of vector projection to relate them to KM and KR.

The vector projection of a vector A on another vector B can be calculated using the formula:

proj_BA = (A · B) / |B|

Where: - · denotes the dot product of two vectors. - |B| denotes the magnitude (length) of vector B.

In our problem, we can find the projections of OM and OR on the plane α using the vectors KM and KR.

Let's denote the projections of OM and OR on the plane α as PM and PR, respectively.

Using the concept of vector projection, we can write:

PM = proj_KMα = (KM · α) / |α|

PR = proj_KRα = (KR · α) / |α|

Since the sum of the lengths of OM and OR is 15 cm, we have:

PM + PR = 15 cm

Substituting the formulas for PM and PR, we get:

[(KM · α) / |α|] + [(KR · α) / |α|] = 15 cm

Now, let's solve this equation to find the value of KR.

To find the distance from point K to plane α, we need to find the length of the projection of the vector KM on the plane α. We can use the formula for PM to calculate it:

PM = (KM · α) / |α|

Finally, let's substitute the value of KR into the formula for PM to find the distance from point K to plane α.

I'll calculate the solution using the given values.