Question

na imagem ao lado, C é o ponto médio do segmento AD, e B é o ponto médio do segmento AC.
Determine as razões:
1) CD/AD
2)AB/AD
3)BD/AC​

Answer 1

1) 1/2

2) 1/4

3) 3

Explicação passo-a-passo:

$\sf AC = \dfrac{AD}{2}$

$\sf CD = AC~~ent\tilde{a}o~~CD = \dfrac{AD}{2}$

Então:

$\sf \dfrac{CD}{AD} = \dfrac{\frac{AD}{2}}{AD}$

$\sf \dfrac{CD}{AD} = \dfrac{AD}{2} \times \dfrac{1}{AD}$

$\sf \dfrac{CD}{AD} = \dfrac{AD}{2 \times AD}$

$\sf \dfrac{CD}{AD} = \Large\red{\boxed{\sf \dfrac{1}{2}}}$

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$\sf AB = \dfrac{AC}{2}~e~AC = \dfrac{AD}{2}$

Então:

$\sf AB = \dfrac{\frac{AD}{2}}{2}$

$\sf AB = \dfrac{AD}{2} \times \dfrac{1}{2}$

$\sf AB = \dfrac{AD}{4}$

Substituindo:

$\sf \dfrac{AB}{AD} = \dfrac{\frac{AD}{4}}{AD}$

$\sf \dfrac{AB}{AD} = \dfrac{AD}{4 \times AD}$

$\sf \dfrac{AB}{AD} =\Large\red{\boxed{\dfrac{1}{4}}}$

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$\sf CD = \dfrac{AD}{2}$

Podemos multiplicar a fração no numerador e denominador por 2, apenas para igualar os denominadores lá na soma que faremos abaixo (BD = BC + CD). Portanto:

$\sf CD = \dfrac{AD}{2} \times \dfrac{2}{2} = \red{\sf \dfrac{2AD}{4}}$

$\sf AC = BC = \red{\sf\dfrac{AD}{4}}$

Então:

$\sf BD = BC + CD$

$\sf BD = \dfrac{AD}{4} + \dfrac{2AD}{4}$

$\sf BD = \dfrac{AD + 2AD}{4}$

$\sf BD = \dfrac{3AD}{4}$

Subsituindo:

$\sf \dfrac{BD}{AC} = \dfrac{\frac{3AD}{4}}{\frac{AD}{4}}$

$\sf \dfrac{BD}{AC} = \dfrac{3AD}{4} \times \dfrac{4}{AD}$

$\sf \dfrac{BD}{AC} = \dfrac{3AD \times 4}{4AD}$

$\sf \dfrac{BD}{AC} = \dfrac{12AD}{4AD}$

$\sf \dfrac{BD}{AC} = \dfrac{12}{4}$

$\sf \dfrac{BD}{AC} =\Large\red{\boxed{\sf 3}}$

Espero que eu tenha ajudado

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