CAT Inequalities in a Triangle - Practice Questions & MCQ

So far, you have been mainly studying the equality of sides and angles of a triangle or triangles. Sometimes, we do come across unequal objects, we need to compare them.

For example,

Line-segment AB is greater in length as compared to line segment CD

is greater than

 

Theorem : If two sides of a triangle are unequal, prove that the angle opposite to the longer side is greater.

Given: .

To Prove:

Construction: Mark a point D on AC such that AD = AB. Join BD

Proof: We know that in a triangle, the angles opposite to equal sides are equal.

Now, in BCD, side CD has been produced to A, forming exterior angle BDA.

 

Theorem : In any triangle, prove that the side opposite to the greater angle is longer.

Given: .

To Prove:

Proof: We have the following possibilities only.

Out of these possibilities, exactly one must be true.

CASE 1:

If possible, let AC = AB.

We know that the angles opposite to equal sides of a triangle are equal.

This contradicts the given hypothesis that .

CASE 2:

If possible, let . Then, .

Since the angle opposite to the longer side is larger, so

This contradicts the given hypothesis that .

CASE 3:

Now, we are left with the only possibility that , which must be true.

Hence, 

Theorem : Prove that, of all the line segments that can be drawn to a given line, from a point not lying on it, the perpendicular line segment is the shortest.

Given: A line AB and a point P outside it. and N is a point, other than M, on AB.

To Prove:

Proof: 

But, in a right-angled triangle, each one of the angles other than the right angle is an acute angle.

But, the side opposite to the smaller angle in a triangle is shorter.

Hence, the perpendicular from P to the given line is shortest of all line segments from P to AB.

Note: The distance between a line and a point, not on it, is the length of perpendicular from the point to the given line.

The distance between a line and a point lying on it, is zero.

 

Theorem : Prove that the sum of any two sides of a triangle is greater than the third side.

Given: .

To Prove:

Construction: Produce BA to D such that AD = AC. Join CD

Proof: 

Theorem : Prove that the difference between any two sides of a triangle is less than its third side.

Given: A line AB and a point P outside it. and N is a point, other than M, on AB.

To Prove:

Construction: Let . Then, along AC, set off AD = AB. Join BD.

Proof: 

Side CD of has been produced to A

 

 

Theorem : Prove that the sum of any two sides of a triangle is greater than twice the median drawn to the third side.

Given:  in which AD is a median.

To Prove: .

Construction: Let AD to E such that AD = DE. Join EC.

Proof: 

We know that the sum of any two sides of a triangle is greater than the third side. So, in , we have