practice problems pending

Q1. The two sides of a triangle are 2 cm and 7 cm. Find the number of possible lengths of the third side. (The length of three sides of the triangle is an integer value)

Solution:
a + 2 > 7
a + 7 > 2
7 + 2 > a
=>
a < 9
a > -5
a > 5
=> 5 < a < 9 => a = 6,7,8
Answer: 3

Q2.

Only using triangle inequality
The three sides of a triangle are 9 cm, 7 cm and 12 cm. Which of the following can be a median of the triangle?

12,14,15,16

Solution:
If the median is falling upon side length 12,then the triangles are:
9,6,x(median length)
7,6,x
Only value of x satisfying both triangle inequalities here is: 12(from the given options).

Similarly,
If the median is falling upon side length 9,then the triangles are:
12,4.5,y(median length)
7,4.5,y
From the given options, none satisfies both.

Similarly,
If the median is falling upon side length 7,then the triangles are:
12,3.5,z(median length)
9,3.5,z
From the given options, 12 satisfies both.
So answer is 12.

Though it's not correct in realty.
Using the Apollonius's Theorem:
median length is 1/2.sqrt(2.a^2 + 2.b^2 - c^2)

which will not come out as 12 for any case.