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1) demonstrate that the bisector of an isosceles triangle is equal to the perpendicular bisector and the height

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Answer to a math question 1) demonstrate that the bisector of an isosceles triangle is equal to the perpendicular bisector and the height

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Para demonstrar que a bissetriz de um triângulo isósceles é igual à bissetriz perpendicular e à altura, podemos seguir estes passos: ### Considere um triângulo isósceles Seja o triângulo \( ABC \) um triângulo isósceles onde \( AB = AC \) e \( BC \) é a base. Seja \( D \) o ponto médio de \( BC \), e seja \( AD \) o segmento de reta do vértice \( A \) ao ponto \( D \). ### Propriedades do Triângulo Isósceles 1. **Bissetor**: O segmento \( AD \) é a bissetriz do ângulo \( \ângulo BAC \). 2. **Mediatriz Perpendicular**: Como \( D \) é o ponto médio de \( BC \), o segmento de reta \( AD \) também é a mediatriz perpendicular de \( BC \). 3. **Altura**: O segmento de reta \( AD \) é a altura do triângulo \( ABC \) do vértice \( A \) à base \( BC \). ### Prova 1. **Mostre que \( AD \) é a Bissetriz**: - Como \( AB = AC \), pelas propriedades dos triângulos isósceles, a bissetriz do ângulo \( AD \) divide \( \ângulo BAC \) em dois ângulos iguais: \[ \ângulo RUIM = \ângulo CAD \] 2. **Mostre que \( AD \) é a Bissetriz Perpendicular**: - Por definição, a mediatriz de um segmento é uma reta que divide o segmento em dois comprimentos iguais e é perpendicular a ele. - Como \( D \) é o ponto médio de \( BC \), temos: \[ BD = CC \] - Para mostrar que \( AD \) é perpendicular a \( BC \), podemos usar o fato de que: \[ \triângulo ABD \cong \triângulo ACD \quad \texto{(por SSS: \( AB = AC \), \( BD = DC \), e \( AD = AD \))} \] - Esta congruência implica que: \[ \ângulo ABD = \ângulo ACD = 90^\circ \] - Assim, \( AD \) é perpendicular a \( BC \). 3. **Mostre que \( AD \) é a Altura**: - A altura de um triângulo é definida como um segmento de reta perpendicular de um vértice à reta que contém o lado oposto. - Como \( AD \) é perpendicular a \( BC \), ele serve como altura do vértice \( A \) até a base \( BC \). ### Conclusão A partir dos passos acima, concluímos que em um triângulo isósceles \( ABC \): - A bissetriz \( AD \) do ângulo \( A \) é igual à bissetriz perpendicular de \( BC \). - O segmento \( AD \) também atua como a altura do vértice \( A \) à base \( BC \). Assim, a bissetriz, a bissetriz perpendicular e a altura são todos o mesmo segmento de reta \( AD \).

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