Q2 of 49 Page 151

Justify the following statements with reasons:

The sum of any two sides of a triangle is greater than twice the median drawn to the third side.

Consider ΔABC and AP is the median as shown BP = PC


Construct PQ such that AP = PQ



Consider ΔAPB and ΔQPC


AP = PQ … construction


∠APB = ∠QCP … vertically opposite angles


BP = PC … AP is median


Thus by SAS test for congruency


ΔAPB ≅ ΔQPC


⇒ AB = CQ … corresponding sides of congruent triangles…(i)


Consider ΔAQC


⇒ AC + CQ > AQ


Using (i)


⇒ AC + AB > AQ


But AQ = AP + PQ and AP = PQ by construction hence AQ = 2AP


⇒ AC + AB > 2AP


More from this chapter

All 49 →
1

Fill in the blanks to make the statements true.

(a) In right triangle the hypotenuse is the ______ side.


(b) The sum of three altitudes of a triangle is ______ than its perimeter.


(c) The sum of any two sides of a triangle is ______ than the third side.


(d) If two angles of a triangle are unequal, then the smaller angle has the ______ side opposite to it.


(e) Difference of any two sides of a triangle is ______ than the third side.


(f) If two sides of a triangle are unequal, then the larger side has ______ angle opposite to it.

 2

Justify the following statements with reasons:

The sum of three sides of a triangle is more than the sum of its altitudes.

 2

Justify the following statements with reasons:

The difference of any two sides of a triangle is less than the third side.

 3

Two triangles ABC and DBC have common base BC. Suppose AB = DC and ∠ABC = ∠BCD. Prove that AC = BD.