Question

Сторона KL параллелограмма KLMN вдвое больше стороны KN. точка P-середина стороны KL.Докажите,что MP-биссектриса угла NML

Answer 1

Прикрепляю.................................

Answer 2

Положим KP = PL = LM = a.Пусть PS || LM, S ∈ MN ⇒ PS = LM = a, MS = LP = a ⇒ PLMS – ромб. MP – диагональ ромба, откуда ∠LMA = ∠SMP ⇒ MP — биссектриса угла NML.

Answer 3

Given Information

We are given that in parallelogram KLMN, side KL is twice the length of side KN, and point P is the midpoint of side KL. We need to prove that MP is the bisector of angle NML.

Proof

To prove that MP is the bisector of angle NML, we need to show that angle NMP is equal to angle PML.

Since KL is twice the length of KN, we can represent KL as 2x and KN as x, where x is a positive real number.

Since P is the midpoint of KL, we can represent KP as x.

Now, let's consider triangle KNP. Since KL is parallel to MN in a parallelogram, angle KNP is equal to angle MNL (alternate interior angles).

Since KP is equal to x and KN is equal to x, triangle KNP is an isosceles triangle.

In an isosceles triangle, the angles opposite the equal sides are equal. Therefore, angle KPN is equal to angle KNP.

Now, let's consider triangle KMP. Since KL is parallel to MN in a parallelogram, angle KMP is equal to angle NML (alternate interior angles).

Since KP is equal to x and KL is equal to 2x, triangle KMP is also an isosceles triangle.

In an isosceles triangle, the angles opposite the equal sides are equal. Therefore, angle KMP is equal to angle KPM.

Since angle KPN is equal to angle KNP and angle KMP is equal to angle KPM, we can conclude that angle NMP is equal to angle PML.

Therefore, MP is the bisector of angle NML.

Conclusion

We have proven that MP is the bisector of angle NML in parallelogram KLMN, given that KL is twice the length of KN and P is the midpoint of KL.