SSC CGL Maths · Triangles

Sides of a Triangle

Any two sides of a triangle add to more than the third, the square on the longest side tells you whether the triangle is right, acute or obtuse, and the altitude to a hypotenuse follows from the area.

All Triangles notes

Why this matters

Sixteen questions in the bank, 13 EASY and 3 MODERATE. Eight ask which lengths can make a triangle, five use Pythagoras or its converse, and three draw the altitude to the hypotenuse. The fast routes are a short list of triples and one counting rule for whole-number sides.

Concept 1 of 3: Triangle inequality: the third side lies between the difference and the sum

Walking straight from A to C is shorter than walking from A to B and then on to C. If the detour were only as long as the straight path, B would sit on the line AC and there would be no triangle.

Definition

Any two sides add to more than the third. If they only equal it, the three points lie on one line.
So when two sides $a$ and $b$ are known, the third side $x$ satisfies $|a - b| < x < a + b$ . Both ends are excluded.
Counting whole-number values: the third side runs from $|a - b| + 1$ to $a + b - 1$ . That is $2 \times$ (the smaller side) $- 1$ values. This shortcut needs whole-number sides.
Each side is less than half the perimeter, because it is less than the sum of the other two.
The longer side faces the larger angle. And the perpendicular is the shortest path from a point to a line, so an altitude is shorter than either side from the same corner.
In an isosceles triangle with two given lengths, try each one as the repeated side and keep the choice that passes the inequality.

Range of the third side

|a - b| < x < a + b \qquad \text{each side} < \frac{a + b + c}{2}

a, bthe two known sides
xthe third side

Worked example

Two sides of a triangle are 9 cm and 14 cm. Between which values must the third side lie, and how many whole-number lengths can it have?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 17 Sep 2024, Shift 3 · Q54Easy

Example 1 · Triangles · Sides of a Triangle

In a triangle, the lengths of sides are 6 units, 12 units and x units. How many possible integer values are there for x?

Including the ends of the range

A third side equal to the sum or to the difference of the other two flattens the triangle into a line, so the range uses strict signs at both ends.

Counting from the difference itself

With sides 4 and 10 the third side runs from 7 to 13, not from 6 to 14. That gives 7 whole-number values, which is 2 × 4 − 1.

Repeating the short side of an isosceles triangle

With sides 4 cm and 9 cm, the triangle 4, 4, 9 fails because 4 + 4 is less than 9. The repeated side must be the 9 cm one.

Drill 7 more on triangle inequality: the third side lies between the difference and the sum

Concept 2 of 3: Pythagoras theorem, triples and the acute or obtuse test

In a right triangle the square on the longest side is exactly the two smaller squares put together. Open the right angle wider and the far side grows, so its square becomes too big; close it and the square becomes too small.

Definition

Right angle at C: $c^2 = a^2 + b^2$ , where $c$ is the hypotenuse, the longest side, opposite the right angle.
Triples to know at sight: 3, 4, 5 · 5, 12, 13 · 8, 15, 17 · 7, 24, 25 · 20, 21, 29 · 9, 40, 41, and any multiple of one (6, 8, 10 or 9, 12, 15).
Converse test, with $c$ the longest side: $c^2 = a^2 + b^2$ right, $c^2 < a^2 + b^2$ acute, $c^2 > a^2 + b^2$ obtuse.
Isosceles triangle: the perpendicular from the apex cuts the base in half, so (half base) $^2$ + height $^2$ = (equal side) $^2$ .
Equilateral triangle of side $a$ : height $= \frac{\sqrt{3}}{2}a$ .
A perpendicular PS onto QR gives two right triangles sharing PS, so $PQ^2 - QS^2 = PS^2 = PR^2 - SR^2$ .

Pythagoras and the equilateral height

c^2 = a^2 + b^2 \qquad h = \frac{\sqrt{3}}{2}a \qquad PQ^2 - QS^2 = PR^2 - SR^2

cthe hypotenuse
hthe height of an equilateral triangle of side a
Sthe foot of the perpendicular from P to QR

Worked example

The two equal sides of an isosceles triangle are 13 cm each and the base is 10 cm. Find the height to the base, and say whether the triangle is acute, right or obtuse.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 18 Sep 2024, Shift 2 · Q56Easy

Example 2 · Triangles · Sides of a Triangle

In a

\Delta PQR

, PQ = PR and PT is perpendicular to QR. If PQ = 17 cm, PT = 15 cm, then what is the measure (in cm) of QR?

Testing with a side that is not the longest

The acute or obtuse test compares the square of the longest side with the sum of the squares of the other two. Any other pairing tells you nothing.

Using the whole base in an isosceles triangle

The perpendicular from the apex splits the base into two equal halves. Put half the base into Pythagoras, then double the result if the base is asked.

Drill 4 more on pythagoras theorem, triples and the acute or obtuse test

Concept 3 of 3: The altitude to the hypotenuse of a right triangle

A right triangle's area can be read two ways: half the product of its legs, or half the hypotenuse times the altitude onto it. Setting the two equal gives the altitude, and Pythagoras in the two smaller triangles gives the pieces of the hypotenuse.

Definition

Let the right angle be at B and BD the altitude to the hypotenuse AC.
Altitude: $\frac{1}{2} \times AB \times BC = \frac{1}{2} \times AC \times BD$ , so $BD = \frac{AB \times BC}{AC}$ .
Each leg squared is its own piece times the hypotenuse: $AB^2 = AD \times AC$ and $BC^2 = DC \times AC$ . The piece AD is the one that touches A.
The altitude squared is the product of the two pieces: $BD^2 = AD \times DC$ .
All of these follow from Pythagoras in triangles ABD and CBD, so use them as a check on each other.
A right triangle with its three corners on a circle has its hypotenuse as a diameter (the angle in a semicircle is 90°).

Altitude to the hypotenuse

BD = \frac{AB \times BC}{AC} \qquad AB^2 = AD \times AC \qquad BD^2 = AD \times DC

Bthe right angle
Dthe foot of the altitude on the hypotenuse AC

Worked example

Triangle ABC is right-angled at B with AB = 15 cm and BC = 20 cm. BD is the altitude to AC. Find BD, AD and DC.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 16 Sep 2025, Shift 1 · Q71Moderate

Example 3 · Triangles · Sides of a Triangle

A right triangle ABC is inscribed in a circle with a diameter of 10 cm. An altitude BD is drawn from the vertex B to the hypotenuse AC. If the length of the leg AB is 6 cm, what is the length of the segment AD?

Taking the radius as the hypotenuse

A right triangle with its corners on a circle has the diameter as its hypotenuse. A circle of radius 5 cm gives a hypotenuse of 10 cm.

Pairing a leg with the far piece

AB² = AD × AC uses the piece AD that touches A. Using DC instead gives BC², the other leg.

Drill 2 more on the altitude to the hypotenuse of a right triangle

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)

Triangle inequality: the third side lies between the difference and the sum
Range of the third side
$|a - b| < x < a + b \qquad \text{each side} < \frac{a + b + c}{2}$
Pythagoras theorem, triples and the acute or obtuse test
Pythagoras and the equilateral height
$c^2 = a^2 + b^2 \qquad h = \frac{\sqrt{3}}{2}a \qquad PQ^2 - QS^2 = PR^2 - SR^2$
The altitude to the hypotenuse of a right triangle
Altitude to the hypotenuse
$BD = \frac{AB \times BC}{AC} \qquad AB^2 = AD \times AC \qquad BD^2 = AD \times DC$

Watch out for (7)

Including the ends of the range
→ Triangle inequality: the third side lies between the difference and the sum
Counting from the difference itself
→ Triangle inequality: the third side lies between the difference and the sum
Repeating the short side of an isosceles triangle
→ Triangle inequality: the third side lies between the difference and the sum
Testing with a side that is not the longest
→ Pythagoras theorem, triples and the acute or obtuse test
Using the whole base in an isosceles triangle
→ Pythagoras theorem, triples and the acute or obtuse test
Taking the radius as the hypotenuse
→ The altitude to the hypotenuse of a right triangle
Pairing a leg with the far piece
→ The altitude to the hypotenuse of a right triangle

Test yourself on Triangles

20 past SSC CGL questions from this chapter, timed at 12 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 16 questions

Related notes

Circles: angles in a circle →