NDA Maths · Lines

Triangles, Quadrilaterals & Polygons

Coordinate geometry applied to figures: the area of a triangle from its vertices, the triangle centres (centroid, incentre, circumcentre), constructing vertices from medians/altitudes, and quadrilateral relations.

All Lines notes NDA Maths strategy

Why this matters

This is the chapter's largest subtopic and its capstone — it combines slope, distance, section and area into figure problems. Knowing the area determinant and the centre formulas cold makes most of it routine.

Concept 1 of 4

Area of a triangle and collinearity

Intuition

The area of a triangle from its three vertices is half the absolute value of a determinant. When that determinant is zero the 'triangle' has collapsed — the points are collinear.

Definition

Area $=\dfrac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$ $=\dfrac12\left|\begin{smallmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{smallmatrix}\right|$ . Collinear iff this is $0$ . The same determinant gives the condition three points lie on a line.

Worked example

Find the area of the triangle with vertices

(0,0),(4,0),(0,3)

Area $=\dfrac12|0(0-3)+4(3-0)+0(0-0)|=\dfrac12|12|$ .

Answer:

6

Practice this conceptself-check · 4 quick reps

Try it yourself

Are

(1,1),(2,3),(3,5)

collinear?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Area of a triangle from vertices?
2.
Collinear ⇒ area?
3.
Area of $(0,0),(4,0),(0,3)$ ?
4.
What test does the area determinant double as?

From the bank · past-year question

Example 1LinesHARD

What is the area of the triangle with vertices

\left(x_1,\frac{1}{x_1}\right), \left(x_2,\frac{1}{x_2}\right), \left(x_3,\frac{1}{x_3}\right)

[Q61 · Sep · 2018]

Drill 4 more on area of a triangle and collinearity

Concept 2 of 4

Centroid, incentre, circumcentre

Intuition

Each triangle centre has a formula: the centroid is the plain average of the vertices; the incentre is the side-length-weighted average; the circumcentre is equidistant from all three vertices (intersection of perpendicular bisectors).

Definition

Centroid: $\left(\dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3}\right)$ .
Incentre: $\dfrac{a\,A+b\,B+c\,C}{a+b+c}$ , with $a,b,c$ the side lengths opposite $A,B,C$ .
Circumcentre: equidistant from all vertices — solve two perpendicular-bisector equations (for a right triangle it is the hypotenuse's midpoint).

Worked example

Find the centroid of the triangle with vertices

(1,2),(3,4),(5,0)

$\left(\dfrac{1+3+5}{3},\dfrac{2+4+0}{3}\right)$ .

Answer:

(3,2)

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the incentre of the equilateral triangle with vertices

A(1,1),B(0,\,\cdots)

where all sides equal — what coincides with what?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Centroid formula?
2.
Incentre weights are?
3.
Circumcentre of a right triangle?
4.
For an equilateral triangle the incentre equals?

From the bank · past-year question

Example 2LinesMODERATE

The incentre of the triangle with vertices

A(1, \sqrt{3})

B(0, 0)

and

C(2, 0)

[Q52 · Apr · 2017]

Drill 2 more on centroid, incentre, circumcentre

Concept 3 of 4

Constructing a triangle: vertices, medians, altitudes

Intuition

Many questions give partial data — midpoints, a median, an altitude — and ask for a vertex or a side's equation. Use midpoint and section relations to recover vertices, and slope/perpendicularity for altitudes and bisectors.

Definition

Recover vertices from midpoints: if $M$ is the midpoint of $BC$ , then $B+C=2M$ . An altitude from a vertex is perpendicular to the opposite side (use negative-reciprocal slope through the vertex). Special triangles: an equilateral/isosceles condition fixes the third vertex (often via rotation or equal-distance). The third vertex of an equilateral triangle on a given base has irrational coordinates in general.

Worked example

Two vertices of a triangle are

B(-5,-1)

and

C(9,3)

. Find the midpoint of

BC

Midpoint $=\left(\dfrac{-5+9}{2},\dfrac{-1+3}{2}\right)$ .

Answer:

(2,1)

Practice this conceptself-check · 4 quick reps

Try it yourself

The midpoint of side

BC

of a triangle is

M(4,2)

and

B=(1,3)

. Find

C

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
If $M$ is the midpoint of $BC$ , then $B+C=$ ?
2.
An altitude is perpendicular to?
3.
Recover $B$ from midpoint $M$ of $AB$ and vertex $A$ ?
4.
Slope of an altitude vs the opposite side?

From the bank · past-year question

Example 3LinesMODERATE

Triangle

ABC

has

A(3,5)

. The mid-points of sides

AB

and

AC

are at

(-1,2)

and

(6,4)

respectively. What are the coordinates of the centroid of the triangle

ABC

[Q62 · Apr · 2024]

Drill 10 more on constructing a triangle: vertices, medians, altitudes

Concept 4 of 4

Parallelograms, squares and diagonals

Intuition

Quadrilateral problems lean on two facts: in a parallelogram the diagonals bisect each other (so the fourth vertex is

D=A+C-B

), and the diagonals' intersection is their common midpoint. Areas come from the cross-product of adjacent side vectors.

Definition

**Parallelogram $ABCD$ :** diagonals bisect each other ⇒ $A+C=B+D$ , so $D=A+C-B$ ; the diagonals meet at the midpoint of either. Area $=|\,\vec{AB}\times\vec{AD}\,|=|x_1y_2-x_2y_1|$ for the side vectors. A square/rectangle from two given parallel sides uses the perpendicular distance for the side length.

Worked example

Three consecutive vertices of a parallelogram are

A(1,2),B(4,3),C(6,6)

. Find

D

$D=A+C-B=(1+6-4,\ 2+6-3)$ .

Answer:

(3,5)

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the area of the parallelogram with

\vec{AB}=(-2,-1)

and

\vec{AD}=(4,3)

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Fourth vertex of parallelogram $ABCD$ ?
2.
Diagonals of a parallelogram do what?
3.
Parallelogram area from side vectors?
4.
Diagonals meet at the ___ of each diagonal.

From the bank · past-year question

Example 4LinesEASY

A parallelogram has three consecutive vertices

(-3, 4)

(0, -4)

and

(5, 2)

. The fourth vertex is

[Q57 · Apr · 2021]

Drill 12 more on parallelograms, squares and diagonals

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1LinesEASY

Let ABC be a triangle. If

D(2,5)

and

E(5,9)

are the mid-points of the sides AB and AC respectively, then what is the length of the side BC?

[Q57 · Apr · 2020]

Example 2LinesMODERATE

If the centroid of a triangle formed by

(7, x)

(y, -6)

and

(9, 10)

(6, 3)

, then the values of

x

and

y

are respectively

[Q60 · Apr · 2017]

Example 3LinesHARD

Direction: Consider the following for the items that follow. The equations of the sides AB, BC and CA of a triangle ABC are x - 2 = 0, y + 1 = 0 and x + 2y - 4 = 0 respectively.

What are the coordinates of circumcentre of the triangle?

[Q58 · Sep · 2021]

Example 4LinesMODERATE

Consider the following for the items that follow: A quadrilateral is formed by the lines

x=0

y=0

x+y=1

and

6x+y=3

What is the equation of other diagonal?

[Q56 · Apr · 2023]

Example 5LinesEASY

What is the maximum number of possible points of intersection of four straight lines and a circle (intersection is between lines as well as between circle and lines)?

[Q37 · Sep · 2024]

Drill every past-year question on this subtopic

32 questions from the bank — paginated, with cart and Word-export support.

Drill the 32 questions

Drill every past-year question on this subtopic

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