Question

mario was given the following enlargement. figures not drawn to scale. what is the length of side a, in centimeters? 12 15 24 33

Explanation:

Step1: Determine the scale factor

Since the figures are similar (enlargement), we can find the scale factor using the corresponding sides. The side of length \(6\) cm in the original figure corresponds to \(12\) cm in the enlarged figure. The scale factor \(k=\frac{12}{6} = 2\).

Step2: Set up a proportion for side \(A\)

Let the length of side \(A\) be \(x\). The corresponding side in the enlarged figure is \(30\) cm. Using the scale factor or proportion \(\frac{x}{30}=\frac{6}{12}\) (cross - multiply). We can also use the relationship between the non - corresponding sides. If we consider the ratio of the other pair of corresponding sides (let's use the ratio of the sides of the original and enlarged trapezoid). The ratio of the height - like sides is \(\frac{6}{12}=\frac{1}{2}\), and for the top - like sides \(\frac{A}{24}\) and for the bottom - like sides \(\frac{A}{30}\) (using the proportion of similar figures). Using the proportion \(\frac{A}{24}=\frac{6}{12}\) (cross - multiply: \(12A=24\times6\)), or \(\frac{A}{30 - 24}=\frac{6}{12 - 6}\) (another way of looking at the trapezoid proportion, but the simplest is \(\frac{A}{24}=\frac{6}{12}\)). Solving \(12A = 24\times6\), we get \(A=\frac{24\times6}{12}\).

Step3: Calculate the value of \(A\)

\(A = 15\) (using the proportion of the sides of similar trapezoids. Let's re - check: If the scale factor from the original to the new trapezoid for the non - parallel sides is \(2\) (since \(6\times2 = 12\)), for the parallel sides, assume the difference between the bottom and top of the original trapezoid is \(y\) and in the new trapezoid is \(30 - 24=6\). Since \(6\times2 = 12\) (non - parallel side scale), but using the ratio of the sides \(\frac{A}{24}=\frac{6 + z}{12+z}\) (where \(z\) is some length, but more simply \(\frac{A}{24}=\frac{30 - A}{30 - 24}\). Cross - multiply: \(6A=24\times(30 - A)\), \(6A=720-24A\), \(6A + 24A=720\), \(30A=720\), \(A = 15\)).

Answer:

15