Triangle MPT with bar(NR) || bar(MT) is shown below. The dimensions are in centimeters.12210107121.jpgWhich measurement is closest to the length of bar(RT) in centimeters?1)1.42)1.83)3.44)4.3

Step 1
Given,
${\displaystyle \mathrm{△}MPT}$ with ${\displaystyle \overline{NR}\mid \mid \overline{MT}}$
PN = 4 cm
PR = 2.5 cm
NM = 2.9 cm
We have to find the length of ${\displaystyle \overline{RT}}$ in centimeters.
Step 2
We have
${\displaystyle \overline{NR}\mid \mid \overline{MT}}$
Then, using the properties for two parallel lines cut by transversal lines
${\displaystyle \mathrm{\angle }PNR=\mathrm{\angle }PMT}$ ..........(1)
${\displaystyle \mathrm{\angle }PRN=\mathrm{\angle }PTM}$ ............(2)
Now,
In ${\displaystyle \mathrm{△}NPR\text{and}\mathrm{△}MPT}$
${\displaystyle \mathrm{\angle }PNR=\mathrm{\angle }PMT}$ (from (1))
${\displaystyle \mathrm{\angle }PRN=\mathrm{\angle }PTM}$ (from (2))
${\displaystyle \mathrm{\angle }NPR=\mathrm{\angle }MPT}$ (common angle )
So,
From AAA (angle-angle-angle) rule of similarity
${\displaystyle \mathrm{△}NPR\sim \mathrm{△}MPT}$
Step 3
Now,
In similar triangle, ratio of corresponding sides will be equal.
${\displaystyle \frac{NP}{MP}=\frac{PR}{PT}=\frac{NR}{MT}}$
Taking
${\displaystyle \frac{NP}{MP}=\frac{PR}{PT}}$
${\displaystyle \Rightarrow \frac{4}{MN+NP}=\frac{2.5}{PR+RT}}$ (substituted the values)
${\displaystyle \Rightarrow \frac{4}{2.9+4}=\frac{2.5}{2.5+RT}}$
${\displaystyle \Rightarrow \frac{4}{6.9}=\frac{2.5}{2.5+RT}}$
${\displaystyle \Rightarrow 4(2.5+RT)=6.9\times 2.5}$
${\displaystyle \Rightarrow 4RT+10=17.25}$
${\displaystyle \Rightarrow 4RT=17.25-10}$
${\displaystyle \Rightarrow 4RT=7.25}$
${\displaystyle \Rightarrow RT=\frac{7.25}{4}}$
${\displaystyle \Rightarrow RT=1.8125}$
So,
The lenght of RT = 1.8cm