Grade 10 Math Formula

1. Linear Relations and Analytic Geometry

Flashcard 1: Slope Formula

• Front: m = \frac{y2 - y1}{x2 - x1}

• Back:

  • Purpose: Calculates the steepness (slope) of a line given two points (x1, y1) and (x2, y2).

  • Variables:

    • m: slope

    • (x1, y1): coordinates of the first point

    • (x2, y2): coordinates of the second point

  • Note: A positive slope means the line goes up from left to right; a negative slope means it goes down.

Flashcard 2: Slope-Intercept Form of a Linear Equation

• Front: y = mx + b

• Back:

  • Purpose: Represents a linear relation, making it easy to identify the slope and y-intercept.

  • Variables:

    • y: dependent variable

    • x: independent variable

    • m: slope of the line

    • b: y-intercept (the point where the line crosses the y-axis, i.e., when x=0)

  • Example: In y = 2x + 3, the slope is 2 and the y-intercept is 3.

Flashcard 3: Standard Form of a Linear Equation

• Front: Ax + By + C = 0

• Back:

  • Purpose: Another common way to represent a linear relation. Often used for systems of equations.

  • Variables:

    • A, B, C: constants (usually integers, with A typically positive)

    • x, y: variables

  • Note: You can convert between standard form and slope-intercept form by rearranging the equation.

Flashcard 4: Distance Formula

• Front: d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

• Back:

  • Purpose: Calculates the distance between two points (x1, y1) and (x2, y2).

  • Variables:

    • d: distance

    • (x1, y1): coordinates of the first point

    • (x2, y2): coordinates of the second point

  • Connection: This formula is derived directly from the Pythagorean theorem.

Flashcard 5: Midpoint Formula

• Front: M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)

• Back:

  • Purpose: Calculates the coordinates of the midpoint of a line segment connecting two points (x1, y1) and (x2, y2).

  • Variables:

    • M: midpoint coordinates

    • (x1, y1): coordinates of the first point

    • (x2, y2): coordinates of the second point

  • Note: It's essentially the average of the x-coordinates and the average of the y-coordinates.

2. Quadratic Relations

Flashcard 6: Standard Form of a Quadratic Equation

• Front: y = ax^2 + bx + c

• Back:

  • Purpose: The most common form of a quadratic relation. The graph is a parabola.

  • Variables:

    • a, b, c: constants (a \neq 0)

    • x, y: variables

  • Note:

    • If a > 0, the parabola opens upwards.

    • If a < 0, the parabola opens downwards.

    • c is the y-intercept.

Flashcard 7: Vertex Form of a Quadratic Equation

• Front: y = a(x - h)^2 + k

• Back:

  • Purpose: Easily identifies the vertex of the parabola.

  • Variables:

    • (h, k): coordinates of the vertex of the parabola

    • a: same 'a' value as in standard form, determines direction of opening and vertical stretch/compression.

  • Note: The axis of symmetry is the vertical line x = h.

Flashcard 8: Factored Form of a Quadratic Equation

• Front: y = a(x - r)(x - s)

• Back:

  • Purpose: Easily identifies the x-intercepts (roots/zeros) of the parabola.

  • Variables:

    • r and s: the x-intercepts (where the parabola crosses the x-axis, i.e., when y=0)

    • a: same 'a' value as in standard form.

  • Note: The x-coordinate of the vertex is the average of the roots: x_v = \frac{r+s}{2}.

Flashcard 9: Quadratic Formula

• Front: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

• Back:

  • Purpose: Solves for the roots (x-intercepts) of any quadratic equation in the form ax^2 + bx + c = 0.

  • Variables:

    • a, b, c: coefficients from the standard form of the quadratic equation.

    • x: the values of x where the parabola crosses the x-axis.

  • Note: This formula can always be used, even when factoring is difficult or impossible.

Flashcard 10: Discriminant

• Front: \Delta = b^2 - 4ac

• Back:

  • Purpose: Determines the number of real roots (x-intercepts) a quadratic equation has without solving the entire quadratic formula.

  • Variables:

    • a, b, c: coefficients from the standard form of the quadratic equation.

  • Interpretation:

    • If \Delta > 0: Two distinct real roots.

    • If \Delta = 0: One real root (a double root).

    • If \Delta < 0: No real roots (the parabola does not intersect the x-axis).

3. Trigonometry (Right Triangles)

Flashcard 11: Pythagorean Theorem

• Front: a^2 + b^2 = c^2

• Back:

  • Purpose: Relates the lengths of the sides of a right-angled triangle.

  • Variables:

    • a, b: lengths of the two shorter sides (legs) of the right triangle.

    • c: length of the longest side (hypotenuse), opposite the right angle.

  • Note: Only applies to right triangles!

Flashcard 12: Sine Ratio (SOH)

• Front: \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}

• Back:

  • Purpose: Relates an angle in a right triangle to the ratio of the length of the side opposite the angle and the hypotenuse.

  • Variables:

    • \theta: the angle in question.

    • "Opposite": length of the side opposite to angle \theta.

    • "Hypotenuse": length of the side opposite the right angle.

Flashcard 13: Cosine Ratio (CAH)

• Front: \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}

• Back:

  • Purpose: Relates an angle in a right triangle to the ratio of the length of the side adjacent to the angle and the hypotenuse.

  • Variables:

    • \theta: the angle in question.

    • "Adjacent": length of the side adjacent to angle \theta (not the hypotenuse).

    • "Hypotenuse": length of the side opposite the right angle.

Flashcard 14: Tangent Ratio (TOA)

• Front: \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

• Back:

  • Purpose: Relates an angle in a right triangle to the ratio of the length of the side opposite the angle and the side adjacent to the angle.

  • Variables:

    • \theta: the angle in question.

    • "Opposite": length of the side opposite to angle \theta.

    • "Adjacent": length of the side adjacent to angle \theta (not the hypotenuse).

4. Measurement and Geometry

Flashcard 15: Area of a Rectangle/Square

• Front: A = l \times w

• Back:

  • Purpose: Calculates the area of a rectangle or square.

  • Variables:

    • A: Area

    • l: length

    • w: width

  • Note: For a square, l=w, so A = s^2 (where s is side length).

Flashcard 16: Area of a Triangle

• Front: A = \frac{1}{2}bh

• Back:

  • Purpose: Calculates the area of any triangle.

  • Variables:

    • A: Area

    • b: length of the base

    • h: perpendicular height from the base to the opposite vertex

Flashcard 17: Area of a Circle

• Front: A = \pi r^2

• Back:

  • Purpose: Calculates the area of a circle.

  • Variables:

    • A: Area

    • \pi: pi (approximately 3.14159)

    • r: radius of the circle

Flashcard 18: Circumference of a Circle

• Front: C = 2\pi r or C = \pi d

• Back:

  • Purpose: Calculates the distance around a circle.

  • Variables:

    • C: Circumference

    • \pi: pi

    • r: radius

    • d: diameter (d = 2r)

Flashcard 19: Volume of a Prism/Cylinder

• Front: V = A_{\text{base}} \times h

• Back:

  • Purpose: Calculates the volume of any prism (e.g., rectangular prism, triangular prism) or cylinder.

  • Variables:

    • V: Volume

    • A_{\text{base}}: Area of the base shape (e.g., l \times w for a rectangular prism, \pi r^2 for a cylinder)

    • h: height of the prism/cylinder

Flashcard 20: Volume of a Pyramid/Cone

• Front: V = \frac{1}{3} A_{\text{base}} \times h

• Back:

  • Purpose: Calculates the volume of any pyramid (e.g., square pyramid, triangular pyramid) or cone.

  • Variables:

    • V: Volume

    • A_{\text{base}}: Area of the base shape (e.g., s^2 for a square pyramid, \pi r^2 for a cone)

    • h: perpendicular height from the base to the apex

Flashcard 21: Volume of a Sphere

• Front: V = \frac{4}{3}\pi r^3

• Back:

  • Purpose: Calculates the volume of a sphere.

  • Variables:

    • V: Volume

    • \pi: pi

    • r: radius of the sphere

Flashcard 22: Surface Area of a Rectangular Prism

• Front: SA = 2(lw + lh + wh)

• Back:

  • Purpose: Calculates the total surface area of a rectangular prism (sum of the areas of all six faces).

  • Variables:

    • SA: Surface Area

    • l: length

    • w: width

    • h: height

Flashcard 23: Surface Area of a Cylinder

• Front: SA = 2\pi r^2 + 2\pi rh

• Back:

  • Purpose: Calculates the total surface area of a cylinder (area of two circular bases plus the area of the curved side).

  • Variables:

    • SA: Surface Area

    • \pi: pi

    • r: radius of the base

    • h: height of the cylinder

Flashcard 24: Surface Area of a Sphere

• Front: SA = 4\pi r^2

• Back:

  • Purpose: Calculates the total surface area of a sphere.

  • Variables:

    • SA: Surface Area

    • \pi: pi

    • r: radius of the sphere