Question

tom has a small triangular-shaped garden in his back yard which measures \\(6\text{ ft} \times 8\text{ ft} \times 10\text{ ft}\\). he wants to extend his garden according to the dashed lines in the below diagram. if he wants the bottom portion, \\(\overline{de}\\), to be \\(20\text{ ft}\\) in length, what lengths would \\(\overline{ad}\\) and \\(\overline{ae}\\) be?

Explanation:

Identify similar triangles

The diagram shows a smaller triangular garden \(\triangle ACB\) with side lengths \(AC = 8\text{ ft}\), \(AB = 6\text{ ft}\), and \(CB = 10\text{ ft}\). The garden is extended along the dashed lines to form a larger triangle \(\triangle ADE\). Since the bottom portion \(\overline{DE}\) is parallel to the original base \(\overline{CB}\), the corresponding angles are equal: \(\angle ADE = \angle ACB\) and \(\angle AED = \angle ABC\). Therefore, by Angle-Angle similarity, the small triangle \(\triangle ACB\) is similar to the large triangle \(\triangle ADE\):

$$\triangle ACB \sim \triangle ADE$$

Determine the scale factor

To find the scale factor between the two similar triangles, we compare the lengths of their corresponding bases, \(\overline{CB}\) and \(\overline{DE}\):

$$\text{Scale Factor} = \frac{DE}{CB} = \frac{20\text{ ft}}{10\text{ ft}} = 2$$

This means every side of the larger triangle \(\triangle ADE\) is twice the length of the corresponding side of the smaller triangle \(\triangle ACB\).

Calculate the length of AD

Using the scale factor, we find the length of the side \(\overline{AD}\) corresponding to side \(\overline{AC}\):

$$AD = 2 \times AC = 2 \times 8\text{ ft} = 16\text{ ft}$$

Calculate the length of AE

Similarly, we find the length of the side \(\overline{AE}\) corresponding to side \(\overline{AB}\):

$$AE = 2 \times AB = 2 \times 6\text{ ft} = 12\text{ ft}$$

Answer:

\(AD = 16\text{ ft}\) and \(AE = 12\text{ ft}\)