Question

triangle abc ~ triangle xyz
triangle abc: sides 9, 15, 18
triangle xyz: sides 6, n, 12
pentagon pqrst ~ pentagon uwxy
pentagon pqrst: sides 6, p, 12, 15, 18
pentagon uwxy: sides 8, 12, 16, 20, k
drag the correct values and scale factors to complete the similarity relationships.
drag the answer choices to the correct category over here!
answer choices:
24
8
3:4
18
10
12
3:2
4:3
2:3
9
value of n
value of p
value of k
scale factor
abc : xyz
scale factor
pqrst : uwxy

Explanation:

Step1: Find the value of \(n\)

Since \(\triangle ABC\sim\triangle XYZ\), the ratios of corresponding sides are equal. \(\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}\). Given \(AB = 8\), \(XY=6\), \(BC = 15\), \(YZ=n\), \(AC = 18\), \(XZ = 12\). Using \(\frac{AB}{XY}=\frac{AC}{XZ}\), \(\frac{8}{6}=\frac{18}{12}\) (check similarity ratio). Also, \(\frac{AB}{XY}=\frac{BC}{YZ}\), so \(\frac{8}{6}=\frac{15}{n}\), cross - multiply: \(8n=6\times15\), \(8n = 90\), \(n=\frac{90}{8}=\frac{45}{4}=11.25\) (wrong, use \(\frac{AB}{XY}=\frac{AC}{XZ}=\frac{8}{6}=\frac{4}{3}\), \(n=\frac{6\times12}{8}=9\) (using \(\frac{XY}{AB}=\frac{XZ}{AC}\), \(\frac{6}{8}=\frac{n}{12}\), cross - multiply \(8n=6\times12\), \(n = 9\))

Step2: Find the value of \(p\)

Since Pentagon \(PQRST\sim\) Pentagon \(UWXY\), the ratios of corresponding sides are equal. Let the scale factor be considered. Using the ratio of corresponding sides. If we take the ratio of two corresponding sides. Let's assume the ratio of sides of \(PQRST\) to \(UWXY\). If we use the first pair of sides: \(\frac{6}{8}\) (side of \(PQRST\) to side of \(UWXY\)). For the side \(p\) and \(12\) (another pair of corresponding sides). \(\frac{6}{8}=\frac{p}{12}\), cross - multiply \(8p=6\times12\), \(p = 9\)

Step3: Find the value of \(k\)

Using the ratio of sides of Pentagon \(PQRST\) to \(UWXY\). \(\frac{6}{8}=\frac{18}{k}\), cross - multiply \(6k=8\times18\), \(k = 24\)

Step4: Find the scale factor \(\triangle ABC:\triangle XYZ\)

The scale factor is the ratio of the sides of \(\triangle ABC\) to \(\triangle XYZ\). \(\frac{AB}{XY}=\frac{8}{6}=\frac{4}{3}\)

Step5: Find the scale factor Pentagon \(PQRST:UWXY\)

The scale factor is the ratio of the sides of Pentagon \(PQRST\) to \(UWXY\). \(\frac{6}{8}=\frac{3}{4}\)

Answer:

Value of \(n = 9\), Value of \(p=9\), Value of \(k = 24\), Scale factor \(\triangle ABC:\triangle XYZ=\frac{4}{3}\), Scale factor Pentagon \(PQRST:UWXY=\frac{3}{4}\)