Coordinate Geometry & Basic Trigonometry: Notes & Examples

BEGINNER

Coordinate Geometry & Basic Trigonometry: visual worked lesson

Core idea

Coordinates locate a point using numbered directions. In a right triangle, trigonometry compares side lengths using an angle.

Solving method

Use coordinates to replace a geometric condition with an equation. In right triangles choose the ratio containing the known side and the required side.

Worked example

Start at (0, 0). Move 3 units right and 2 units up. Where are you? 1. The first coordinate records the horizontal move. 2. The second coordinate records the vertical move. 3. Your new position is (3, 2).

Read the diagram

Start at the intersection of the axes. The horizontal arrow travels three units right, and the vertical arrow travels two units up. The point is (3, 2), with x written first. The arrows also form the legs of a right triangle; relative to an angle, trigonometric ratios compare its sides.

Exam-style application

Point P=(x,0) is equidistant from A=(1,2) and B=(5,4). Squared distances give (x-1)^2+4=(x-5)^2+16. Expand and cancel x^2: -2x+5=-10x+41, so 8x=36 and x=4.5. Squared distances avoid unnecessary square roots. Check both values are equal before accepting P.

Common mistakes

Order matters: (3, 2) and (2, 3) usually describe different points. The distance formula uses squared coordinate differences; the slope formula does not. Keep these separate.

Coordinates record horizontal and vertical moves. Move 3 units right, then 2 units up; equal axis scales are used. — Coordinates record horizontal and vertical moves

Move 3 units right, then 2 units up; equal axis scales are used. — Coordinates record horizontal and vertical moves

Formulas to remember

Distance between points = √[(x₂ − x₁)² + (y₂ − y₁)²]

sin θ = opposite ÷ hypotenuse

Source record: Bharatiya Pathshala — original illustrated lessons.

BEGINNER

Core idea

Coordinates locate a point on a grid. In (x, y), x tells you how far to move sideways and y tells you how far to move vertically. Trigonometry connects angles with side lengths in right triangles. The hypotenuse is the longest side, opposite the right angle.

BEGINNER

Steps

1. Plot the points carefully. 2. Count the horizontal and vertical changes. 3. For distance, use a right triangle and the Pythagorean rule. 4. For slope, divide vertical change by horizontal change. 5. In trigonometry, label opposite, adjacent, and hypotenuse relative to the chosen angle.

BEGINNER

Worked example

From (0, 0) to (3, 4), the horizontal change is 3 and the vertical change is 4. The straight distance is the square root of 3 squared + 4 squared, which is the square root of 25 = 5.

BEGINNER

Common mistake

A vertical line has an undefined slope because its horizontal change is zero. Do not divide by zero. Opposite and adjacent sides depend on which angle you are using.

Basic Trigonometry

In a right triangle, sine is opposite/hypotenuse and cosine is adjacent/hypotenuse. The chosen acute angle determines opposite and adjacent.

Worked example

A ladder 10 m long makes 30 degrees with the ground. Height = 10 sin(30) = 5 m. The ladder, not the vertical height, is the hypotenuse.

Application and strategy

Use coordinates to replace a geometric condition with an equation. In right triangles choose the ratio containing the known side and the required side.

Worked application

Point P=(x,0) is equidistant from A=(1,2) and B=(5,4). Squared distances give (x-1)^2+4=(x-5)^2+16. Expand and cancel x^2: -2x+5=-10x+41, so 8x=36 and x=4.5. Squared distances avoid unnecessary square roots. Check both values are equal before accepting P.

Common traps and strategy

The distance formula uses squared coordinate differences; the slope formula does not. Keep these separate.

Coordinates, lines, distances and circles

Core idea

A point (x,y) gives horizontal and vertical coordinates. Distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²]. The midpoint is ((x₁+x₂)/2,(y₁+y₂)/2).

Solving method

A nonvertical line has slope m=(y₂−y₁)/(x₂−x₁). Point-slope form is y−y₁=m(x−x₁); slope-intercept form is y=mx+c. Vertical lines have equation x=k and undefined slope. Parallel nonvertical lines have equal slopes; perpendicular nonvertical lines have slopes whose product is −1. A horizontal and a vertical line are also perpendicular. The internal division point with AP:PB=m:n is ((nx₁+mx₂)/(m+n),(ny₁+my₂)/(m+n)). A circle with centre (h,k) and radius r satisfies (x−h)²+(y−k)²=r². Distance from (x₀,y₀) to Ax+By+C=0 is |Ax₀+By₀+C|/√(A²+B²), with A and B not both zero.

Worked example

Between (1,2) and (4,6), distance is √(3²+4²)=5, midpoint is (2.5,4), and slope is 4/3. A line through (1,2) with that slope is y−2=(4/3)(x−1).

Common mistakes

Do not divide by zero for vertical slopes. Similar-looking equations may use different axes or units. The area of a triangle is half the absolute determinant of its three coordinate rows, so a zero determinant means collinear points.

Formulas to remember

Distance = √[(x₂−x₁)²+(y₂−y₁)²]

Slope = (y₂−y₁)/(x₂−x₁) when x₂≠x₁

Source record: Bharatiya Pathshala — reviewed preparation lessons.

Coordinate Geometry & Basic Trigonometry, explained simply

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