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Solution: Since, in DABC, AM is a median from vertex A to the opposite side BC.
We know that a median from a vertex divides the opposite side in two equal parts.
Therefore, BM = MC ….(i)
In DABM, AB + BM > AM …(ii)
Similarly, in DAMC, MC + AC > AM …(iii)
Adding (ii) and (iii), we get
AB + BM + MC + AC > AM + AM
AB + (BM + MC) + AC > AM + AM
AB + BC + AC > 2AM (Q BC = BM + MC)
Practice Time 8B
1. State in which of the following cases, you can form a triangle with the following given sides:
(a) 5 cm, 6 cm, 4 cm (b) 3.2 cm, 1.4 cm, 1.8 cm
(c) 2.8 cm, 3.4 cm, 4.8 cm (d) 1.2 cm, 2.2 cm, 4.6 cm A
2. D is any point on the side BC of the DABC.
Prove that AB + BC + CA > 2AD.
3. P is a point in the interior of DABC. B D C
State whether the following statements are ‘True’ or ‘False’. A
(a) AP + PB > AB (b) AB + BP < AP
(c) PB + PC > BC (d) BC + PC > PB P
(e) PA + PC > AC (f) AC + AP > PC
B C
4. The lengths of two sides of a triangle are 6 cm and 9 cm. Between what two measures should the
length of the third side fall?
5. Find all possible integer values for the third side of a triangle if the measures of two sides are 7 cm
and 9 cm.
Pythagoras Property of a Right Angled Triangle
We know that a triangle having an angle of measure 90° is called a B
right-angled triangle. The sides in a right-angled triangle are given
special names. The longest side opposite to the right angle is called Hypotenuse
hypotenuse, and the other two sides are known as the legs of the Leg
right-angled triangle, they are also called base and perpendicular a c
respectively.
In a right angle triangle ABC, right angle at C, AB is the opposite side Right angle
of right angle C, so AB is hypotenuse. And BC and AC are the legs of C b Leg A
the DABC, or the perpendicular and the base, respectively.
A right angle triangle has a special property called Pythagoras property.
According this property, the square of the hypotenuse is equal to the sum of the squares of other two sides.
This property is also known as Pythagoras Theorem. Let us understand Pythagoras property
through an activity.
181 The Triangle and Its Properties
