Answer :
The length of line QP is [tex]\sqrt{14}[/tex]
What are similar triangles?
Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.
ΔPOR and ΔPNO are similar since line ON and line QR are parallel.
for similar triangles, the ratio of corresponding lengths is equal.
So that,
QR/ON = QP/NP = PR/OP
but ON = QP
since QR/ON = QP/NP
then
QR/QP = QP/NP
by cross multiplying we have
QP^2 = NP x RQ
but QR =2 and NP = 7
QP^2 = 2 x 7
QP^2 = 14
QP = [tex]\sqrt{14}[/tex]
In conclusion, QP = [tex]\sqrt{14}[/tex]
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