Inequalities of Triangle

Triangles are three-sided closed figures and show a variance in properties depending on the measurement of sides and angles. From equilaterals to scalene triangles, we come across a variety of triangles, yet while studying triangle inequality we need to keep in mind some properties that let us study the variance. In the chapter below we shall throw light on the many properties that determine the triangle inequality.

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Inequalities of Triangle

In the previous chapter, we have studied the equality of sides and angles between two triangles or in a triangle. There may be instances when we come across unequal objects and this is when we start comparing them to reach to conclusions.

In our instances of comparisons, we take into consideration every part of the object. Likewise, when we compare various parts of a triangle, we compare its every part individually. For instance, take the line segment shown in the figure:

We can see that line segment AB is smaller than line segment PQ. The difference is visually clear, yet to know the difference and affirm our findings we measure both of them. After measuring both, we reach the conclusion which was visually clear. Likewise, see the figure of angles below:

The angles AOB and POQ are unequal. We can clearly see that ∠ POQ is greater than ∠AOB. These figures of unequal line segments and unequal angles have a close relationship between unequal sides and unequal angles of a triangle. Let’s see how:

1. If two sides of a triangle are unequal, the angle opposite to the longer side is greater than others.

For proving this point we need to do an activity. On a drawing board fix three pins at points PQR. A line segment is drawn at PQ. Taking P as center and some radius we draw an arc at A. Similarly with different radius we draw a few more arcs at point B, C, and D.

On joining these points with P as well as B we observe that as we move from A to D, the ∠P is becoming larger with every arc. Now, what happens to the side opposite to the angle. We observe here that the length of the side is also increasing.

Let’s take another triangle, which seems scalene in appearance. A scalene has all sides of different length. On measuring the length of the sides in a scalene triangle, we come to a conclusion that angle opposite to the longer side is the greatest while the angle opposite to the shortest side is the smallest.

2. The side opposite to the larger angle is longer, in any triangle.

Now, let’s draw a triangle with all three unequal sides. Measure each side of the triangle. In this triangle when we measure with a protractor, we find that the side opposite to the largest angle is the longest as compared to the other two sides.

3. The sum of any two sides of a triangle is always greater than the third side.

Take a triangle XYZ with the measurement of its sides known to you. Find the sum of sides XY + YZ, YZ+ZX and ZX+XY separately. What do you observe, in all the three cases, the sum of two sides is greater than the third side. From the above set of properties of lines and angles, we can easily understand the basis of various triangle inequality.

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Solved Example for You on Triangle Inequality

X is a point on side QR of ΔPQR such that PX=PR. Show that PQ>PX

Solution: The figure shown above is of ΔPQR, with a point X on the line QR We join P with X to form a line segment PX. In ΔXPR, we see that PX = PR, which is given in the question. This gives us that ∠PXR = ∠PRX because angles opposite to equal sides are also equal.

Now, ∠PXR is an exterior angle for ΔPQX. So we have that ∠PXR > ∠PQX This can also be written as
∠PRX > ∠PQX
or, ∠PRQ > ∠PQR
So, PQ>PR as the side opposite to larger angle is longer.
or, PQ>PX