Sum of any Two Sides is Greater than Twice the Median

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Sum of any Two Sides is Greater than Twice the Median

Theorem Statement:
Prove that sum of any two sides of a triangle is greater than twice the median?

Proof:
Given Triangle ABC in which AD is the median.
To prove: AB + AC > 2AD
Construction :
median
Extend AD to E such that AD = DE.
Now join EC.
Proof:
In ΔADB and ΔEDC
AD=DE[ By construction]
D is the midpoint BC.[DB=DB]
ΔADB=ΔEDC [vertically opposite angles]
Therefore Δ ADB ≅ ΔEDC [ By SAS congruence criterion.]
AB=ED[Corresponding parts of congruent triangles ]
In ΔAEC,
AC+ED> AE [sum of any two sides of a triangle is greater than the third side]
AC+AB>2AD[AE=AD+DE=AD+AD=2AD and ED=AB]
Hence proved.

Do Check:

FAQs on the sum of any two sides is Greater than Twice the Median

1. What is the relationship between the 2 parts of the median?

The centroid is always two-thirds of the way along each median from that median’s interior angle. This means that it sets up a 2:1 for each of the three medians: 2 parts of the median are between the centroid and the interior angle. 1 part of the median is between the centroid and the opposite side.

2. What is the relation between the median and sides of a triangle?

The relation between the median and the sides of a triangle is such that three times the sum of squares of the length of sides is equal to 4 times the squares of medians of a triangle.

3. What is the difference between median and altitude?

The difference between medians and altitudes is that a median is drawn from a vertex of the triangle to the midpoint of the opposite side, whereas an altitude is drawn from a vertex of the triangle to the opposite side being perpendicular to it.

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