Geometry — Grade X

Grade X

GEOMETRY || ज्यामिति

Geometry ज्यामिति

Math — Grade X · Chapter 5

📚 3 Topics

🎯 TQ 8 · TM 13

🔢 K:2 · U:2 · A:2 · HA:2

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Geometry

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Exercise — Model 1

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Exercise — Model 2

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Exercise — Model 3

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Key Formulas

Geometry — essential formulas

Area of Triangle — Base & Height

$$A = \frac{1}{2} \times b \times h$$

$b$ = base, $h$ = perpendicular height from base to opposite vertex.

Area of Triangle — Heron's Formula

$$A = \sqrt{s(s-a)(s-b)(s-c)}$$

$s = \frac{a+b+c}{2}$ is the semi-perimeter. Use when all three sides are known.

Area of Triangle — Two Sides & Angle

$$A = \frac{1}{2}ab\sin C$$

$a$, $b$ = two sides, $C$ = included angle between them.

Area of Parallelogram

$$A = b \times h$$

$b$ = base, $h$ = perpendicular height. A rectangle is a special parallelogram.

Area of Trapezium

$$A = \frac{1}{2}(a + b) \times h$$

$a$, $b$ = parallel sides, $h$ = perpendicular distance between them.

Area of Rhombus

$$A = \frac{1}{2} \times d_1 \times d_2$$

$d_1$, $d_2$ = the two diagonals of the rhombus.

Circle — Circumference

$$C = 2\pi r = \pi d$$

$r$ = radius, $d$ = diameter. Use $\pi \approx 3.14159$.

Circle — Area

$$A = \pi r^2$$

Area enclosed by a circle of radius $r$.

Arc Length

$$\ell = \frac{\theta}{360°} \times 2\pi r$$

$\theta$ = central angle in degrees. Fraction of the full circumference.

Area of Sector

$$A = \frac{\theta}{360°} \times \pi r^2$$

$\theta$ = central angle. A sector is a "pie slice" of the circle.

Inscribed Angle Theorem

$$\angle \text{inscribed} = \frac{1}{2} \angle \text{central}$$

An inscribed angle is half the central angle subtending the same arc.

Tangent–Radius

$$OT \perp PT$$

The radius to a point of tangency is always perpendicular to the tangent line.

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Key Concepts

Understand beyond memorising formulas

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Area of Triangle

Choose the right formula based on what is given: use ½bh when base and height are known, Heron's when all three sides are given, and ½ab sin C when two sides and the included angle are given.

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Area of Quadrilateral

Split any quadrilateral into two triangles by drawing a diagonal, then sum the triangle areas. Special cases: parallelogram uses $b \times h$, trapezium uses ½(a+b)h, rhombus uses ½d₁d₂.

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Geometric Construction

Constructions use only compass and straightedge. Key constructions include: bisecting a line/angle, constructing perpendiculars, constructing triangles given SSS/SAS/ASA, and constructing similar/parallel lines.

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Circle — Key Terms

Know the difference: chord (line inside circle), secant (line crossing circle), tangent (touches at one point), arc (part of circumference), sector (pie slice), segment (chord + arc region).

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Circle Theorems

Key theorems: (1) Angle at centre = 2 × inscribed angle. (2) Angles in the same segment are equal. (3) Opposite angles of a cyclic quadrilateral add to 180°. (4) Tangent ⊥ radius at point of contact.

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Tangent Properties

Two tangents drawn from the same external point are equal in length. The line joining the external point to the centre bisects the angle between the two tangents and is the perpendicular bisector of the chord joining the two points of tangency.

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