CDS Mathematics · Triangles

Pythagoras Theorem

In a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides, and any triangle whose sides satisfy that is right-angled.

All Triangles notes CDS Mathematics strategy

Why this matters

Twenty-two PYQs, four of them HARD. They come in three kinds: integer triples, the perimeter-and-area pair (always solved by squaring a + b), and word problems — ladders, poles, walks — where the work is drawing the right triangle. Knowing six triples by sight saves most of the arithmetic.

Concept 1 of 3: Pythagorean triples

Most exam triangles have whole-number sides, and there are only a few small triples. Recognise

3, 4, 5

5, 12, 13

inside a question — scaled or not — and the arithmetic disappears.

Definition

Know by sight: $3, 4, 5$ · $5, 12, 13$ · $8, 15, 17$ · $7, 24, 25$ · $20, 21, 29$ · $9, 40, 41$ , and their multiples ( $6, 8, 10$ ; $9, 12, 15$ ; $15, 20, 25$ ).
Generator: $m^2 - n^2$ , $2mn$ , $m^2 + n^2$ is a triple for any $m > n$ .
Odd leg $n$ : $n, \dfrac{n^2 - 1}{2}, \dfrac{n^2 + 1}{2}$ is a triple (so $11, 60, 61$ ).
The hypotenuse is always the largest side, so put the largest expression on the right.

Triple generator

(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2

Worked example

The sides of a right triangle are three consecutive whole numbers. Find them.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q63Hard

Example 1 · Triangles · Pythagoras Theorem and its Converse

If one side of a right-angled triangle (with all sides integers) is 15 cm, then what is the maximum perimeter of the triangle ?

A ratio fixes only the shape

'Sides in the ratio

x : (x - 1) : (x - 18)

' strictly allows any multiple. The paper means the sides ARE those expressions; solve for

x

and reject any root that makes a side negative.

Drill 7 more on pythagorean triples

Concept 2 of 3: Perimeter and area together

Two facts about the legs — their sum and their product — are all you need, because squaring the sum gives the hypotenuse's square plus twice the product. The area gives the product; the perimeter gives the sum once the hypotenuse is taken out.

Definition

With legs $a, b$ , hypotenuse $c$ , area $\Delta = \dfrac{ab}{2}$ and perimeter $P$ :

$(a + b)^2 = c^2 + 2ab = c^2 + 4\Delta$ ;
$(a - b)^2 = c^2 - 4\Delta$ ;
since $a + b = P - c$ : $(P - c)^2 = c^2 + 4\Delta$ , which gives $c = \dfrac{P^2 - 4\Delta}{2P}$ .

Hypotenuse from perimeter and area

(a + b)^2 = c^2 + 4\Delta, \qquad c = \dfrac{P^2 - 4\Delta}{2P}

Worked example

A right triangle has perimeter

30

and area

30

. Find its hypotenuse.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (II) 2025 — Elementary Mathematics · Q67Moderate

Example 2 · Triangles · Pythagoras Theorem and its Converse

ABC

is a triangle right angled at

B

. Further,

(AB + BC)

exceeds

AC

by 10 units. If the perimeter of the triangle is 60 units, then what is the area of the triangle?

The squares of ALL three sides

a^2 + b^2 + c^2 = 2c^2

, not

c^2

. A question giving 'the sum of the squares of the sides' wants

c = \sqrt{\text{sum} \div 2}

Drill 5 more on perimeter and area together

Concept 3 of 3: Ladders, poles and walks

Every word problem hides one right triangle. Find it: the ladder is the hypotenuse against a wall; two poles give a horizontal gap and a height difference; a walk gives a net east and a net north.

Definition

Ladder: length $L$ , foot $x$ from the wall, top at height $h$ : $L^2 = x^2 + h^2$ . Sliding changes $x$ and $h$ but not $L$ .
Two poles: tips are $\sqrt{d^2 + (h_1 - h_2)^2}$ apart, where $d$ is the gap between them.
Walks: add the east–west moves and the north–south moves separately, then combine.
Equilateral triangle of side $a$ : height $\dfrac{\sqrt3}{2}a$ , so $h^2 = 3\left(\dfrac a2\right)^2$ .

Pythagoras

c^2 = a^2 + b^2

Worked example

10

m ladder reaches

8

m up a wall. Its foot slides

2

m further from the wall. How far does the top slide down?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q86Moderate

Example 3 · Triangles · Pythagoras Theorem and its Converse

A ladder is resting against a vertical wall and its bottom is 2.5 m away from the wall. If it slips 0.8 m down the wall, then its bottom will move away from the wall by 1.4 m. What is the length of the ladder ?

Poles use the DIFFERENCE of heights

The vertical leg between two pole tips is

h_1 - h_2

, not either height. That is also why 'heights differ by

10

m' is enough information on its own.

Drill 7 more on ladders, poles and walks

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)

Pythagorean triples
Triple generator
$(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2$
Perimeter and area together
Hypotenuse from perimeter and area
$(a + b)^2 = c^2 + 4\Delta, \qquad c = \dfrac{P^2 - 4\Delta}{2P}$
Ladders, poles and walks
Pythagoras
$c^2 = a^2 + b^2$

Watch out for (3)

A ratio fixes only the shape
→ Pythagorean triples
The squares of ALL three sides
→ Perimeter and area together
Poles use the DIFFERENCE of heights
→ Ladders, poles and walks

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 22 questions