Std. 9
Mathematics

≤ and ≥ include the endpoint; < and > exclude it.

The line joining the mid-points of two sides of a triangle is parallel to the third side and half its length.

A line through the mid-point of one side parallel to another side bisects the third side.

Parallel-line angle relationships lead to similar triangles and proportional sides.

DE∥BC and AD=DB ⇒ AE=EC in the corresponding triangle configuration.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. c²=a²+b²

The side opposite the right angle is the hypotenuse and is the longest side.

If one side squared equals the sum of the other two squares, the opposite angle is 90°.

Identify the right angle before applying the theorem.

A quadrilateral has four sides and its interior angles total 360°. Sum=360°

Opposite sides are parallel and equal; opposite angles are equal; diagonals bisect each other.

A rectangle has four right angles. A rhombus has four equal sides.

A square has four equal sides and four right angles. A trapezium has one pair of parallel sides in the usual ICSE treatment.

Geometrical constructions use ruler and compass to create figures accurately from given conditions.

A regular polygon has equal sides and equal interior angles.

Use sharp pencil, clean arcs and accurate intersections; construction lines show the mathematical method.

Every arc and line should establish one of the required geometric conditions.

Triangles on the same base and between the same parallels have equal areas.

On the same base and between the same parallels, a triangle has half the area of the parallelogram. Area triangle=½bh

Equal base and equal height explain the area relationship.

Area theorems allow equal areas to be proved without calculating dimensions.

A circle is the set of points at a fixed distance from a fixed centre. That distance is the radius.

A chord joins two points. The diameter is the longest chord and passes through the centre.

The angle at the centre standing on an arc is twice the angle at the circumference standing on the same arc: ∠centre=2∠circumference.

Opposite angles of a cyclic quadrilateral are supplementary: ∠A+∠C=180°.

Statistics involves collecting, organising, presenting and interpreting numerical information.

Frequency tells how many times a value or class occurs.

Class limits, class boundaries, class size and frequency must be identified correctly.

Bar graphs, histograms and frequency polygons present data visually; histograms use adjoining rectangles for continuous intervals.

The arithmetic mean is Mean=Σx/n.

Arrange ungrouped data. For odd n, take the middle observation; for even n, average the two middle observations.

Mean is sensitive to unusually large or small observations; median is generally less affected.

Count the observations before selecting the median position.

Perimeter is the total length around a plane figure.

s=(a+b+c)/2 Area=√[s(s−a)(s−b)(s−c)]

C=2πr A=πr²

Split complicated figures into familiar shapes and add or subtract areas as required.

Surface area measures exposed outer area; volume measures three-dimensional space occupied.

TSA=2(lb+bh+hl) V=lbh

TSA=6a² V=a³

An open box has fewer exposed faces. Distinguish internal and external dimensions in thickness problems.

For an acute angle, trigonometric ratios compare opposite, adjacent and hypotenuse sides.

sinθ=O/H cosθ=A/H tanθ=O/A

cosecθ=1/sinθ secθ=1/cosθ cotθ=1/tanθ

Identify the reference angle and label O, A and H before choosing a ratio.

Know exact values for 0°, 30°, 45°, 60° and 90°.

sin30°=1/2 cos60°=1/2 tan45°=1

sinθ=cos(90°−θ) tanθ=cot(90°−θ)

Use exact fractions and surds when evaluating expressions rather than unnecessary decimal approximations.

Solve a right triangle by combining Pythagoras and trigonometric ratios when sufficient information is known.

Draw and label the hidden right triangle before selecting a formula.

The opposite and adjacent sides depend on the chosen angle.

Confirm the side lengths and angles are physically and geometrically possible.

Two angles are complementary when their sum is 90°. A+B=90°

sinA=cos(90°−A) cosA=sin(90°−A)

tanA=cot(90°−A) secA=cosec(90°−A)

Complementary identities simplify expressions and connect the two acute angles of a right triangle.

The x-axis and y-axis are perpendicular and meet at the origin.

A point is written (x,y): x gives horizontal position and y gives vertical position.

The independent variable is selected or controlled; the dependent variable changes in response.

Move x units horizontally and y units vertically, using signs to locate the quadrant.

Each linear equation represents a straight line; the common solution is their intersection.

Plot both equations on the same axes and read the intersection coordinates.

Intersecting lines give one solution; parallel distinct lines give none; coincident lines give infinitely many common points.

Substitute the intersection coordinates into both equations.

Horizontal and vertical differences form a right triangle, so Pythagoras gives the distance.

d=√[(x₂−x₁)²+(y₂−y₁)²]

Reversing the two points does not change the distance because the differences are squared.

Use the formula for straight-line distance between two coordinate points.

Cost price is the amount paid; selling price is received; marked price is the stated price before discount.

Profit=SP−CP Loss=CP−SP

Profit%=Profit/CP×100 Loss%=Loss/CP×100

Discount=MP−SP Discount%=Discount/MP×100

Construct triangles accurately from suitable side and angle information using ruler and compass.

Draw one side and locate the third vertex by arcs with the other two given side lengths.

Construct the included angle and then mark the second given side.

Use the given angle-side conditions exactly and check the resulting triangle.

Rearrange an equation so the required variable is alone.

A=πr² → r=√(A/π)

Clear denominators when that simplifies the rearrangement.

Substitute the new expression into the original formula.

Similar figures have the same shape but can have different sizes. Corresponding angles are equal and corresponding sides are proportional.

Triangle similarity can be established by AA, SAS or SSS conditions where applicable.

k=corresponding length/corresponding length

Similarity helps determine unknown lengths and establish proportional relationships.