Question

на боковой стороне CT трапеции ABCT отмечена такая точка M что AB=AT=AM=CT, докажите что треугольник BCM равнобедренный​

Answer 1

Пошаговое объяснение:

Дано: АВСТ - трапеция

AB=AT=AM=CT

Доказать: ΔВСМ - равнобедренный.

Доказательство:

1. Рассмотрим АВСТ.

АВ = СТ (условие)

⇒ АВСТ - равнобедренная трапеция.

• В равнобедренной трапеции углы при основаниях равны.

⇒ ∠А = ∠Т.

• В трапеции углы, прилежащие к боковой стороне равны в сумме 180°.

⇒ ∠В = 180° - ∠А

2. Рассмотрим ΔАМТ.

АТ = АМ (условие)

⇒ ΔАМТ - равнобедренный.

• Углы при основании равнобедренного треугольника равны.

⇒ ∠Т = ∠1

3.     Сумма смежных углов равна 180°.

∠АМС = 180°-∠1 (смежные)

 ∠В     = 180° - ∠А (п.1)

∠А = ∠Т = ∠1 (п.1,2)

⇒ ∠В = ∠АМС

4. Рассмотрим ΔАВМ.

АВ = АМ (условие)

⇒ ΔАВМ - равнобедренный.

⇒ ∠2 = ∠3 (при основании р/б треугольника)

5. Рассмотрим ΔВСМ.

∠МВС = ∠В - ∠3

∠ВМС = ∠АМС-∠2

⇒ ∠МВС = ∠МВС

• Если в треугольнике равны два угла, то этот треугольник равнобедренный.

⇒ ΔВСМ - равнобедренный.

Answer 2

Given Information

We are given a trapezoid ABCD, where AB = AT = AM = CT, and M is a point on the lateral side of the trapezoid.

To Prove

We need to prove that triangle BCM is isosceles (or, in other words, that BM = CM).

Proof

To prove that triangle BCM is isosceles, we will use the given information and properties of trapezoids.

Since AB = AT = AM = CT, we can conclude that triangle ABM and triangle CMT are isosceles triangles. This is because in an isosceles triangle, two sides are equal in length.

Now, let's consider triangle ABM. Since AB = AM, we can conclude that angle ABM = angle AMB. This is because in an isosceles triangle, the angles opposite the equal sides are also equal.

Similarly, in triangle CMT, since CT = CM, we can conclude that angle CMT = angle CTM.

Now, let's consider the trapezoid ABCD. The sum of the angles in a trapezoid is 360 degrees. Therefore, angle ABM + angle AMB + angle CMT + angle CTM = 360 degrees.

Substituting the equal angles we found earlier, we have angle ABM + angle ABM + angle CMT + angle CMT = 360 degrees.

Simplifying, we have 2 * angle ABM + 2 * angle CMT = 360 degrees.

Dividing both sides by 2, we have angle ABM + angle CMT = 180 degrees.

Since angle ABM = angle CMT (as proved earlier), we can rewrite the equation as 2 * angle ABM = 180 degrees.

Dividing both sides by 2, we have angle ABM = 90 degrees.

Now, let's consider triangle BCM. We have angle ABM = 90 degrees. Since angle ABM is an angle in triangle BCM, we can conclude that triangle BCM is a right triangle.

In a right triangle, the hypotenuse is always the longest side. Therefore, in triangle BCM, BM is the hypotenuse and BC is one of the legs.

Since BM is the longest side, we can conclude that BM > BC.

However, in an isosceles triangle, the legs are equal in length. Therefore, if triangle BCM is isosceles, BM must be equal to BC.

Since we have shown that BM > BC, we can conclude that triangle BCM is not isosceles.

Therefore, we have proved that the triangle BCM is not isosceles.

Conclusion

Based on the given information and the properties of trapezoids, we have proved that the triangle BCM is not isosceles.