Question

1. in addition to the information given in the drawing, which statement would be sufficient to that \\( \triangle stw\sim\triangle xyz\\)?

a. \\( tw:sw = 1:3\\)
b. \\( tw:sw = 8:5\\)
c. \\( sw = 24\\) and \\( tw = 15\\)
d. \\( sw = 15\\) and \\( tw = 24\\)

Explanation:

Step1: Recall the Side - Side - Side (SSS) similarity criterion

For two triangles \(\triangle STW\) and \(\triangle XYZ\) to be similar by SSS similarity (\(\triangle A\sim\triangle B\) if \(\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}\)), we need the ratios of their corresponding sides to be equal. The sides of \(\triangle STW\) are \(ST = 27\), and if we assume the correspondence \(ST\) corresponds to \(XY=9\), \(SW\) corresponds to \(XZ = 5\), and \(TW\) corresponds to \(YZ=8\).

Step2: Calculate the ratio of \(ST\) to \(XY\)

The ratio of \(ST\) to \(XY\) is \(\frac{ST}{XY}=\frac{27}{9}=3\). So, we need \(\frac{SW}{XZ}=\frac{TW}{YZ}=3\) or \(\frac{TW}{SW}=\frac{YZ}{XZ}\)

If \(\frac{TW}{SW}=\frac{8}{5}\) (from \(YZ = 8\) and \(XZ = 5\) in \(\triangle XYZ\)), when \(\frac{TW}{SW}=\frac{8}{5}\), and \(\frac{ST}{XY}=\frac{27}{9} = 3\), by SSS similarity (if we assume the angles between the sides are equal, but since the problem is about the ratio of sides for similarity condition based on the given options).

Let's check each option:

• Option A: \(\frac{TW}{SW}=\frac{1}{3}\), not equal to \(\frac{8}{5}\)
• Option B: \(\frac{TW}{SW}=\frac{8}{5}\), which is the ratio of \(YZ\) to \(XZ\) in \(\triangle XYZ\)
• Option C: \(\frac{TW}{SW}=\frac{15}{24}=\frac{5}{8}

eq\frac{8}{5}\)

• Option D: \(\frac{TW}{SW}=\frac{24}{15}=\frac{8}{5}\), but we need to check the ratio with \(ST\) and \(XY\). But if \(\frac{TW}{SW}=\frac{8}{5}\) (the ratio of two sides of \(\triangle XYZ\)), and assuming the angle between \(ST - SW\) and \(XY - XZ\) is common (by the problem's intention of using side - ratio for similarity as per the given options in a multiple - choice format where SSS or SAS (if angle is implicit) similarity is considered. In the context of the problem, since we are given side - length ratios in the options and a pair of sides (\(ST = 27\) and \(XY = 9\)) with a ratio of \(3\), and for \(\triangle STW\sim\triangle XYZ\) by SSS (if we assume the correspondence \(ST - XY\), \(SW - XZ\), \(TW - YZ\)), the ratio of \(TW\) to \(SW\) should be the same as the ratio of \(YZ\) to \(XZ\))

Answer:

B. \(TW:SW = 8:5\)