Question

На прямой MN между точками M и N выбрана точка А и проведены по одну сторону от MN лучи АВ, АС и АD. На луче АВ выбрана точка К и через неё проведена прямая, параллельная MN и пересекающая лучи АС и АD соответственно в точках Р и Е, КР=РА=РЕ. Докажите, что АВ перпендикулярно АD.

Answer 1

.к. ab||mk то треугольники ABN и MNK подобны т.к. O - точка пересечения медиан, она делит ее в соотношении 2:1 AB=2/3MK MK=1,5*12=18

Answer 2

Given Information

We are given that on the line segment MN, a point A is chosen. Rays AB, AC, and AD are drawn on one side of MN. On ray AB, a point K is chosen, and a line parallel to MN is drawn through K, intersecting rays AC and AD at points P and E respectively. It is also given that KR = RA = RE.

To Prove

We need to prove that AB is perpendicular to AD.

Proof

To prove that AB is perpendicular to AD, we can use the fact that if two lines are perpendicular, then the product of their slopes is -1.

Let's assume the coordinates of points M, N, A, B, D, K, P, and E as (x₁, y₁), (x₂, y₂), (x₃, y₃), (x₄, y₄), (x₅, y₅), (x₆, y₆), (x₇, y₇), and (x₈, y₈) respectively.

The slope of line MN can be calculated using the formula:

m₁ = (y₂ - y₁) / (x₂ - x₁)

Similarly, the slope of line AB can be calculated using the formula:

m₂ = (y₄ - y₃) / (x₄ - x₃)

Since line AB is parallel to line MN, the slopes of AB and MN are equal. Therefore, we have:

m₁ = m₂

Next, let's calculate the slope of line AD using the formula:

m₃ = (y₅ - y₃) / (x₅ - x₃)

Since line AD is perpendicular to line AB, the product of their slopes is -1. Therefore, we have:

m₂ * m₃ = -1

Substituting the values of m₂ and m₃, we get:

[(y₄ - y₃) / (x₄ - x₃)] * [(y₅ - y₃) / (x₅ - x₃)] = -1

Simplifying the equation, we have:

[(y₄ - y₃) * (y₅ - y₃)] = -[(x₄ - x₃) * (x₅ - x₃)]

Now, let's consider the points K, R, and A. We are given that KR = RA. Therefore, the coordinates of points K, R, and A satisfy the equation:

[(x₆ - x₄) * (x₃ - x₆)] + [(y₆ - y₄) * (y₃ - y₆)] = 0

Since KR = RA, we can rewrite the equation as:

[(x₃ - x₄) * (x₆ - x₃)] + [(y₃ - y₄) * (y₆ - y₃)] = 0

Expanding the equation, we have:

[(x₃ * x₆) - (x₄ * x₆) - (x₃ * x₃) + (x₄ * x₃)] + [(y₃ * y₆) - (y₄ * y₆) - (y₃ * y₃) + (y₄ * y₃)] = 0

Simplifying the equation, we get:

[(x₃ * x₆) - (x₃ * x₃)] + [(y₃ * y₆) - (y₃ * y₃)] = [(x₄ * x₆) - (x₄ * x₃)] + [(y₄ * y₆) - (y₄ * y₃)]

Further simplifying the equation, we have:

x₃ * (x₆ - x₃) + y₃ * (y₆ - y₃) = x₄ * (x₆ - x₃) + y₄ * (y₆ - y₃)

Comparing this equation with the equation we obtained earlier, we can see that:

[(x₆ - x₃) * (x₅ - x₃)] + [(y₆ - y₃) * (y₅ - y₃)] = [(x₄ - x₃) * (x₅ - x₃)] + [(y₄ - y₃) * (y₅ - y₃)]

Therefore, we can conclude that:

m₃ = m₂

Since the slopes of lines AB and AD are equal, and the product of their slopes is -1, we can conclude that AB is perpendicular to AD.

Hence, the statement "AB is perpendicular to AD" is proven.

Note: The proof assumes that the given points are not collinear. If the points are collinear, the proof may not hold.