Question

b. what is the length of segment ( bb )?

1. explain how you know that segment ( de ) is not parallel to segment ( bc ).
2. in right triangle ( abc, ac = 4 ) and ( bc = 5 ). a new triangle ( dec ) is formed by connecting the midpoints of ( ac ) and ( bc ).

a. what is the area of triangle ( abc )?
b. what is the area of triangle ( dec )?
c. does the scale factor for the side lengths apply to the area as well?

Explanation:

Step1: Calculate the area of triangle \(ABC\)

The formula for the area of a right - triangle is \(A=\frac{1}{2}\times base\times height\).
For right - triangle \(ABC\) with \(AC = 4\) (height) and \(BC = 5\) (base), we have \(A_{ABC}=\frac{1}{2}\times AC\times BC\).
Substitute \(AC = 4\) and \(BC = 5\) into the formula: \(A_{ABC}=\frac{1}{2}\times4\times5\).

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Step2: Calculate the area of triangle \(DEC\)

Since \(D\) is the mid - point of \(AC\), \(DC=\frac{1}{2}AC\). Given \(AC = 4\), then \(DC = 2\).
Since \(E\) is the mid - point of \(BC\), \(EC=\frac{1}{2}BC\). Given \(BC = 5\), then \(EC = 2.5\).
The area of right - triangle \(DEC\) is \(A_{DEC}=\frac{1}{2}\times DC\times EC\).
Substitute \(DC = 2\) and \(EC = 2.5\) into the formula: \(A_{DEC}=\frac{1}{2}\times2\times2.5\).

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Step3: Analyze the scale factor and area relationship

The scale factor of the side lengths of \(\triangle DEC\) to \(\triangle ABC\) is \(k=\frac{1}{2}\).
The ratio of the areas is \(\frac{A_{DEC}}{A_{ABC}}=\frac{2.5}{10}=\frac{1}{4}\).
Since \((\frac{1}{2})^2=\frac{1}{4}\), the scale factor for the side lengths (\(k = \frac{1}{2}\)) when squared gives the ratio of the areas.

Answer:

a. The area of triangle \(ABC\) is \(10\).
b. The area of triangle \(DEC\) is \(2.5\).
c. Yes, the scale factor for the side lengths (if \(k\) is the scale factor of side lengths) when squared (\(k^{2}\)) gives the ratio of the areas of the two similar triangles.