Algebra Class 10 - Linear Equations and Quadratics

A pair of linear equations in two variables can be represented graphically by lines. The lines can intersect, be coincident, or be parallel, each indicating a different solution scenario.

When two lines intersect, they have exactly one point in common, representing a unique solution to the system of equations.

If two lines are coincident (overlap completely), they share all points and have infinitely many solutions, indicating that the equations are dependent.

Parallel lines never intersect, indicating that the system of equations has no solution. The lines are consistent but have no common points.

The general form of a quadratic equation is ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. This form is crucial for understanding and solving quadratic equations.

The roots of a quadratic equation are the values of x that satisfy the equation. These roots are the same as the zeroes of the quadratic polynomial.

The quadratic formula is a solution for finding the roots of a quadratic equation. It uses the coefficients of the equation to directly calculate the two possible roots. The formula is derived from completing the square of the general quadratic equation.

The discriminant of a quadratic equation is the expression D = b² - 4ac. It determines the nature of the roots (real or complex) and the number of real roots (one or two).

Factoring a quadratic equation involves expressing it as a product of two linear factors. This method can be applied to find the roots of the equation by setting each factor to zero.

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

This criterion states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, then the triangles are similar.

If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the triangle is right-angled.

The areas of two similar triangles are proportional to the squares of their corresponding sides (Area of ∆ АВС / Area of A DEF = AB² / DE²).

A system that uses ordered pairs of numbers (x, y) to represent points in a plane. It consists of two perpendicular lines, the x-axis and the y-axis.

The point where the x-axis and y-axis intersect, with coordinates (0, 0).

The horizontal distance of a point from the y-axis. It is the first number in the ordered pair (x, y).

A line that touches a circle at exactly one point. It's a special case of a secant where the two end points of its corresponding chord coincide.

The angle formed between the line of sight and a horizontal line when looking down at an object.

The angle formed between the line of sight and a horizontal line when looking up at an object.

The tangent at any point on a circle is perpendicular to the radius drawn to that point.

If two tangents are drawn from an external point to a circle, then the lengths of these tangents are equal.

The distance from the center of a circle to any point on the circle.

A collection of all points in a plane that are equidistant from a fixed point.

A line that intersects a circle at two distinct points.

The nth term of an arithmetic progression (AP) is found by adding the first term ('a') to the product of (n-1) and the common difference ('d').

The sum of the first 'n' terms of an arithmetic progression (AP) is calculated using the formula: Sn = n/2 [2a + (n-1)d], where 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms.

The sum of the first 'n' natural numbers is obtained by multiplying 'n' with (n + 1) and then dividing by 2.

Two shapes are congruent if they have the same size and shape. This means corresponding sides and angles are equal.

Two shapes are similar if they have the same shape but different sizes. This means their corresponding angles are equal, and their sides are in proportion.

If three points are on a line, the ratio of the lengths of the segments formed by the line is equal to the ratio of the corresponding segments formed by a parallel line.

Two triangles are similar if their corresponding angles are equal. This is also known as the Angle-Angle-Angle (AAA) Similarity criterion.

Two triangles are similar if their corresponding sides are proportional. This is known as the Side-Side-Side (SSS) Similarity criterion.

In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact. This means that the line segment connecting the points where the chord touches the larger circle is divided in half by the point where it touches the smaller circle.

If two tangents are drawn from an external point to a circle, the angle formed between the two tangents is twice the angle formed between the point of contact and the center of the circle.

Tangents drawn at the ends of a diameter of a circle are always parallel to each other. This means that the two lines never intersect, no matter how far they are extended.

If a quadrilateral is drawn so that all its sides are tangent to a circle, then the sum of the lengths of opposite sides of the quadrilateral are equal.

The angle formed between two tangents drawn from an external point to a circle is supplementary (adds up to 180 degrees) to the angle formed by connecting the points of contact and the center of the circle.

The distance around a circle, calculated by the formula C = 2πr where r is the radius.

The flat space enclosed inside a circle, calculated by the formula A = πr² where r is the radius.

The distance across a circle, passing through the center. It is twice the radius.

The space enclosed by a portion of a circle's boundary and two radii. It's like a slice of pizza.

The distance along the curved part of a sector of a circle. It's part of the circle's circumference.

The space enclosed by a part of a circle's boundary and a chord. It's like a pizza slice minus the triangle formed by the crust.

A 3-dimensional rectangular solid with six faces, twelve edges, and eight vertices. It's like a box.