Question

1. 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 \(A=\frac{1}{2}\times base\times height\).
In right - triangle \(ABC\), \(AC = 4\) (height) and \(BC = 5\) (base).
So, \(A_{ABC}=\frac{1}{2}\times AC\times BC=\frac{1}{2}\times4\times5\).

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Step2: Calculate the side - lengths of triangle \(DEC\)

Since \(D\) is the mid - point of \(AC\) and \(E\) is the mid - point of \(BC\), \(DC=\frac{1}{2}AC\) and \(EC=\frac{1}{2}BC\).
Given \(AC = 4\) and \(BC = 5\), then \(DC=\frac{1}{2}\times4 = 2\) and \(EC=\frac{1}{2}\times5=\frac{5}{2}\).

Step3: Calculate the area of triangle \(DEC\)

Using the area formula for a right - triangle \(A=\frac{1}{2}\times base\times height\) for \(\triangle DEC\) with \(DC = 2\) (height) and \(EC=\frac{5}{2}\) (base).
\(A_{DEC}=\frac{1}{2}\times DC\times EC=\frac{1}{2}\times2\times\frac{5}{2}\).

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Step4: 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 \(\frac{A_{DEC}}{A_{ABC}}=\frac{\frac{5}{2}}{10}=\frac{1}{4}\), and \(k^{2}=(\frac{1}{2})^{2}=\frac{1}{4}\).

Answer:

a. The area of triangle \(ABC\) is \(10\).
b. The area of triangle \(DEC\) is \(2.5\).
c. Yes, if the scale - factor of side - lengths is \(k\), the scale - factor of the areas is \(k^{2}\). Here, the side - length scale - factor \(k = \frac{1}{2}\) and the area scale - factor \(k^{2}=\frac{1}{4}\) (since \(\frac{2.5}{10}=\frac{1}{4}\)).