Question

with the information given below, determine how the triangles can be shown to be similar.

m = 28 cm
n = 14 cm
p = 20 cm
q = 14 cm
r = 7 cm
s = 10 cm

note: figures not drawn to scale.

a. the triangles are similar by sss.
b. the triangles are similar by sas.
c. the triangles are not similar.
d. the triangles are similar by aa.

Explanation:

Step1: Calculate the ratios of corresponding sides

For the sides \(n = 14\) cm and \(q = 14\) cm, \(r = 7\) cm and \(n = 14\) cm, \(s = 10\) cm and \(p = 20\) cm.
The ratio of \(n\) to \(q\) is \(\frac{n}{q}=\frac{14}{14} = 1\).
The ratio of \(r\) to \(n\) is \(\frac{r}{n}=\frac{7}{14}=\frac{1}{2}\).
The ratio of \(s\) to \(p\) is \(\frac{s}{p}=\frac{10}{20}=\frac{1}{2}\).
Since \(\frac{r}{n}=\frac{s}{p}
eq\frac{n}{q}\), the SSS (Side - Side - Side) similarity criterion (which requires \(\frac{a}{a'}=\frac{b}{b'}=\frac{c}{c'}\)) is not met.

Step2: Check for SAS similarity

The SAS (Side - Angle - Side) similarity criterion requires two sides in proportion and the included angle equal. But we are not given any information about the angles. However, if we consider the ratios of the sides:
Let's assume the sides are \(n\) and \(p\) in one triangle and \(q\) and \(s\) in the other. \(\frac{n}{q}=\frac{14}{14} = 1\), \(\frac{p}{s}=\frac{20}{10}=2\). Also, \(\frac{m}{q}=\frac{28}{14} = 2\), \(\frac{n}{r}=\frac{14}{7}=2\), \(\frac{p}{s}=\frac{20}{10}=2\)
We have \(\frac{m}{q}=\frac{n}{r}=\frac{p}{s}=2\)

Answer:

A. The triangles are similar by SSS