Prove the midpoint theorem. In the given triangle $NO\parallel LM$, KN=2.7cm, KL=5.4cm, KO=3.9cm, Find OM.
Hint: Here in this question we will first proof mid-point theorem and with the help of its result further we will solve the numerical part of the question. Mid-point theorem: - It says a line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.
Complete step-by-step answer: Draw a $\vartriangle ABC$ where E and F are the mid-points of side AB and AC. Through point ‘C’ draw a line segment parallel to AB and extend EF to meet this line at point ‘D‘
Since $AB\parallel CD$ (by construction) and ED is a transversal line then from the property of parallel lines we can say that $\angle AEF = \angle CDF$ (Alternate angles) ...........................equation (1) In $\vartriangle AEF$ and $\vartriangle CDF$ $\angle AEF = \angle CDF$ (From equation (1)) $\angle AFE = \angle CFD$ (Vertically opposite angle) $AF = FC$ (As F is the midpoint of AC) [By AAS (angle angle side) congruence rule] So, EA=CD (By CPCT) But EB=EA (Because E is the mid-point of AB) Therefore EB=CD Now in EBCD, $EB\parallel DC\& EB = DC$ (Proved above) Thus one pair of opposite sides is equal and parallel. Hence EBCD is a parallelogram. Since opposite sides of parallelograms are parallel. So, $ED\parallel BC$ $\therefore EF\parallel BC$ (As F is a point on line ED) Thus a mid-point theorem which states that a line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side is proved. Now we will solve another part of the question: -
In the given triangle $NO\parallel LM$, KN=2.7cm, KL=5.4cm, KO=3.9cm and we have to find OM. As $NO\parallel LM$ we can say through mid-point theorem that N and O are the mid-points of side KL and KM Therefore KN=NL and KO=OM So, NL=2.7cm and OM=3.9cm Hence final answer is OM=3.9cm
Note: Students may likely make mistakes while applying the mid-point theorem they should be cautious about which sides are parallel and which points are mid-points related to these parallel sides.
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