Question

triangle qrs and its dimensions are shown. which measurements in centimeters represent the dimensions of a triangle that is similar to triangle qrs? 14 cm, 20 cm, 23 cm 3 cm, 6 cm, 7.5 cm 8 cm, 16 cm, 20 cm 1 cm, 2 cm, 3 cm

Explanation:

Step1: Find the ratio of sides

For triangle \(QRS\), the sides are \(6\mathrm{cm}\), \(12\mathrm{cm}\), \(15\mathrm{cm}\).
Check the ratios for each option.
For the option \(3\mathrm{cm},6\mathrm{cm},7.5\mathrm{cm}\):
\(\frac{3}{6}=\frac{1}{2}\), \(\frac{6}{12}=\frac{1}{2}\), \(\frac{7.5}{15}=\frac{1}{2}\)

Step2: Check triangle inequality for other options

• For \(1\mathrm{cm},2\mathrm{cm},3\mathrm{cm}\): \(1 + 2=3\), does not satisfy triangle inequality (\(a + b>c\) for all sides \(a,b,c\) of a triangle)
• For \(8\mathrm{cm},16\mathrm{cm},20\mathrm{cm}\): \(\frac{8}{6}=\frac{4}{3}\), \(\frac{16}{12}=\frac{4}{3}\), \(\frac{20}{15}=\frac{4}{3}\), but \(8 + 16=24>20\), \(8+20 = 28>16\), \(16 + 20=36>8\). But we need to check similarity ratio consistency. The ratio of sides of original triangle \(QRS\) is \(6:12:15 = 2:4:5\). For \(8:16:20=2:4:5\) but wait, no, \(6:12:15=\frac{6}{3}:\frac{12}{3}:\frac{15}{3}=2:4:5\), \(8:16:20=\frac{8}{4}:\frac{16}{4}:\frac{20}{4}=2:4:5\). But wait, no, the ratio of similarity. Let's use the formula for similar triangles. If \(\triangle ABC\sim\triangle DEF\), then \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\). For \(3,6,7.5\) and \(6,12,15\), \(\frac{3}{6}=\frac{6}{12}=\frac{7.5}{15}=\frac{1}{2}\). For \(8,16,20\) and \(6,12,15\), \(\frac{8}{6}=\frac{4}{3}\), \(\frac{16}{12}=\frac{4}{3}\), \(\frac{20}{15}=\frac{4}{3}\), but \(6\times\frac{4}{3}=8\), \(12\times\frac{4}{3} = 16\), \(15\times\frac{4}{3}=20\). But also check triangle inequality for \(1,2,3\): \(1+2 = 3\) (fails). For \(14,20,23\): \(\frac{14}{6}=\frac{7}{3}\), \(\frac{20}{12}=\frac{5}{3}\), \(\frac{23}{15}\) (ratios not equal)

Answer:

\(3\mathrm{cm},6\mathrm{cm},7.5\mathrm{cm}\)