Question

△abc and △def are similar.
(a) there are three proportions below. the ratios in each proportion compare the length of one of the sides of △abc to the length of the corresponding side of △def. complete the proportions.
\frac{ab}{de}=square \frac{ac}{square}=square \frac{bc}{square}=\frac{2}{3}
(b) choose the correct statement about the answers to part (a).
○ each pair of side lengths compared is not in the same ratio. this is because in a proportion both ratios must be different.
○ each pair of side lengths compared is not in the same ratio. this is because the triangles are not right triangles.
○ each pair of side lengths compared is in the same ratio. this is coincidence. we would usually not expect this from similar triangles that are not the same size.
○ each pair of side lengths compared is in the same ratio. this is because the side lengths in each pair are lengths of corresponding sides and the triangles are similar.

Explanation:

Paso 1: Calcular la proporción \(\frac{AB}{DE}\)

Dado que \(AB = 6\) y \(DE=9\), entonces \(\frac{AB}{DE}=\frac{6}{9}=\frac{2}{3}\)

Paso 2: Completar la proporción \(\frac{AC}{?}\)

Como \(\triangle ABC\sim\triangle DEF\), los lados correspondientes son proporcionales. \(AC = 8\), y el lado correspondiente en \(\triangle DEF\) es \(DF = 12\). Entonces \(\frac{AC}{DF}=\frac{8}{12}=\frac{2}{3}\)

Paso 3: Completar la proporción \(\frac{BC}{?}\)

\(BC = 10\), y el lado correspondiente en \(\triangle DEF\) es \(EF = 15\). Entonces \(\frac{BC}{EF}=\frac{10}{15}=\frac{2}{3}\)

Paso 4: Analizar la parte (b)

Como \(\triangle ABC\sim\triangle DEF\), por la definición de triángulos similares, los lados correspondientes están en la misma proporción.

Respuesta:

(a) \(\frac{AB}{DE}=\frac{2}{3}\), \(\frac{AC}{DF}=\frac{2}{3}\), \(\frac{BC}{EF}=\frac{2}{3}\)

(b) \( \text{Each pair of side lengths compared is in the same ratio. This is because the side lengths in each pair are lengths of corresponding sides and the triangles are similar.}\)

Answer:

Paso 1: Calcular la proporción \(\frac{AB}{DE}\)

Dado que \(AB = 6\) y \(DE=9\), entonces \(\frac{AB}{DE}=\frac{6}{9}=\frac{2}{3}\)

Paso 2: Completar la proporción \(\frac{AC}{?}\)

Como \(\triangle ABC\sim\triangle DEF\), los lados correspondientes son proporcionales. \(AC = 8\), y el lado correspondiente en \(\triangle DEF\) es \(DF = 12\). Entonces \(\frac{AC}{DF}=\frac{8}{12}=\frac{2}{3}\)

Paso 3: Completar la proporción \(\frac{BC}{?}\)

\(BC = 10\), y el lado correspondiente en \(\triangle DEF\) es \(EF = 15\). Entonces \(\frac{BC}{EF}=\frac{10}{15}=\frac{2}{3}\)

Paso 4: Analizar la parte (b)

Como \(\triangle ABC\sim\triangle DEF\), por la definición de triángulos similares, los lados correspondientes están en la misma proporción.

Respuesta:

(a) \(\frac{AB}{DE}=\frac{2}{3}\), \(\frac{AC}{DF}=\frac{2}{3}\), \(\frac{BC}{EF}=\frac{2}{3}\)

(b) \( \text{Each pair of side lengths compared is in the same ratio. This is because the side lengths in each pair are lengths of corresponding sides and the triangles are similar.}\)