Q2 of 49 Page 151

Justify the following statements with reasons:

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

Consider ΔABC with altitudes as AP, BQ and CR on segments BC, AB and AC respectively



Consider ΔAPC


∠APC = 90° … AP is altitude


∠ACP is some acute angle less than 90°


Hence AC > AP … (i)


Similarly we can prove


BC > CR … (ii)


AB > BQ … (iii)


Adding (i), (ii) and (iii) we get


⇒ AC + BC + AB > AP + CR + BQ


⇒ sum of sides > sum of altitudes


More from this chapter

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5

Let ABC be a triangle and P be an interior point. Prove that AB + BC + CA < 2 (PA + PB + PC).

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

 2

Justify the following statements with reasons:

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