Question

use the given information to prove that ( np = qp ).

given: ( n ) is the midpoint of ( overline{mp} ).
( p ) is the midpoint of ( overline{qr} ). ( mn = pr ).
prove: ( np = qp )

Explanation:

Step1: Use the mid - point definition

Since \(N\) is the mid - point of \(\overline{MP}\), by the definition of a mid - point (\(MN = NP\)). Since \(P\) is the mid - point of \(\overline{QR}\), by the definition of a mid - point (\(QP=PR\)).

Step2: Substitute using the given equality

We are given that \(MN = PR\). Substituting \(MN\) with \(NP\) (from \(MN = NP\)) and \(PR\) with \(QP\) (from \(QP = PR\)) in the equation \(MN = PR\), we get \(NP=QP\).

Answer:

\(MN = NP\) (because \(N\) is the mid - point of \(\overline{MP}\)) and \(QP = PR\) (because \(P\) is the mid - point of \(\overline{QR}\))