CDS Mathematics · Triangles

Triangle Inequalities

Any two sides together are longer than the third, the longer side faces the larger angle, and the three medians together lie between three-quarters of the perimeter and the whole perimeter.

All Triangles notes CDS Mathematics strategy

Why this matters

Ten PYQs, and half of them are HARD, the highest share of any page in the chapter. They are almost all statement questions: which inequality always holds. Three facts settle every one of them, and the traps are inequalities printed the wrong way round.

Concept 1 of 3: The triangle inequality

The straight path between two corners is the shortest, so going round by the third corner is always longer. If two sides only just add up to the third, the triangle has collapsed into a straight line.

Definition

For sides $a, b, c$ :

$a + b > c$ , $b + c > a$ and $c + a > b$ ;
equivalently, each side lies strictly between the difference and the sum of the other two: $|b - c| < a < b + c$ .

To test three lengths, add the two smaller and compare with the largest. Since $a - b - c < 0$ for every side, a product like $(a - b - c)(b - c - a)(c - a - b)$ is a product of three negatives, so it is negative.

Triangle inequality

|b - c| < a < b + c

Worked example

Which of

(5, 6, 12)

(4, 5, 9)

and

(6, 8, 13)

can be the sides of a triangle?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q70Easy

Example 1 · Triangles · Triangle Inequalities

Which one of the following triples does not represent the sides of a triangle ?

Equal is not enough

4 + 5 = 9

gives a flat line, not a triangle. The inequality is strict.

When two options are both true

If a quantity is always positive, 'non-negative' is also true. CDS has printed both; mark the tighter description, 'positive', which is the one the setter means.

Drill 3 more on the triangle inequality

Concept 2 of 3: The larger side faces the larger angle

Open the angle between two fixed sides and the third side grows. So the order of the sides is the order of the angles opposite them, and the side facing a right or obtuse angle is the longest.

Definition

In any triangle, $a > b$ exactly when $A > B$ .
The hypotenuse faces the right angle, so it is the longest side.
If $c$ is the longest side: $c^2 < a^2 + b^2$ means the triangle is acute, $=$ means right, $>$ means obtuse.
In a right triangle $a^3 + b^3 = a\cdot a^2 + b\cdot b^2 < c(a^2 + b^2) = c^3$ , since each leg is shorter than $c$ .

Acute, right or obtuse

c^2 \lessgtr a^2 + b^2 \quad (c \text{ the longest side})

Worked example

In triangle

ABC

A = 50^\circ

and

B = 60^\circ

. Put the sides in order.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q30Moderate

Example 2 · Triangles · Triangle Inequalities

Let the bisector of the angle

BAC

of a triangle

ABC

meet

BC

X

. Which one of the following is correct ?

Compare the angles, not the picture

A figure that is not drawn to scale can make any side look the longest. Name the angles in the small triangle that holds the two sides, then compare the angles opposite them; the larger angle faces the longer side.

Drill 1 more on the larger side faces the larger angle

Concept 3 of 3: The medians against the perimeter

A median joins a vertex to the midpoint of the opposite side. Double it into a parallelogram and the triangle inequality says the two sides beside it are longer than twice the median. Apply the triangle inequality at the centroid instead and the medians turn out longer than three-quarters of the perimeter.

Definition

Let $m_a, m_b, m_c$ be the medians and $P = a + b + c$ .

$b + c > 2m_a$ , and likewise for the other two.
Adding the three: $m_a + m_b + m_c < P$ .
At the centroid $G$ , $GB + GC > a$ gives $\dfrac23(m_b + m_c) > a$ ; adding the three: $m_a + m_b + m_c > \dfrac34 P$ .
Also, for any point $D$ on $BC$ : $AB + BC + CA > 2AD$ .

Sum of the medians

\tfrac34 (a + b + c) < m_a + m_b + m_c < a + b + c

Worked example

Check the bounds on an equilateral triangle of side

2

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q90Hard

Example 3 · Triangles · Triangle Inequalities

Consider the following for the next two (02) questions : In a triangle ABC, a, b and c are the lengths of the sides and p, q and r are the lengths of its medians.

Which one of the following is correct ?

Read which way the inequality points

Statement questions print the true result reversed: 'the sum of two sides is less than twice the median'. The true fact is greater. Check any printed inequality on an equilateral triangle, where each median is

\dfrac{\sqrt3}{2}

of the side.

Drill 3 more on the medians against the perimeter

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

The triangle inequality
Triangle inequality
$|b - c| < a < b + c$
The larger side faces the larger angle
Acute, right or obtuse
$c^2 \lessgtr a^2 + b^2 \quad (c \text{ the longest side})$
The medians against the perimeter
Sum of the medians
$\tfrac34 (a + b + c) < m_a + m_b + m_c < a + b + c$

Watch out for (4)

Equal is not enough
→ The triangle inequality
When two options are both true
→ The triangle inequality
Compare the angles, not the picture
→ The larger side faces the larger angle
Read which way the inequality points
→ The medians against the perimeter

Test yourself on Triangles

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Triangles chapter test Drill all 10 questions