Probability - Probability Calculator
Calculate probabilities for different experiments
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Applications of Trigonometry – Height and Distance
Applications of Trigonometry - Height and Distance
Solve real-world problems using trigonometry
Height of an Object
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Trigonometric Calculator
Trigonometry - Trigonometric Calculator
Calculate trigonometric ratios and solve right triangles
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Mensuration – 3D Shapes Calculator
Mensuration - 3D Shapes Calculator
Calculate surface area and volume of 3D shapes
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Tangents and Secants – Circle Geometry
Tangents and Secants - Circle Geometry
Explore properties of tangents and secants to circles
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Triangle Similarity Checker
Similar Triangles - Triangle Similarity Checker
Check if two triangles are similar using different criteria
Triangle 1
Triangle 2
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Coordinate Geometry – Distance Calculator
Coordinate Geometry - Distance Calculator
Calculate distances and relationships between points
Point A
Point B
Point C
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Progressions – Sequence Calculator
Progressions - Sequence Calculator
Calculate arithmetic and geometric progressions
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Quadratic Equations – Quadratic Solver
Quadratic Equations - Quadratic Solver
Solve quadratic equations of the form ax² + bx + c = 0
x² +
x +
= 0
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Pair of Linear Equations – Equation Solver
Pair of Linear Equations - Equation Solver
Solve a pair of linear equations in two variables.
x +
y =
x +
y =
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Polynomial Operations
Polynomials - Polynomial Operations
Enter coefficients for two polynomials and perform operations.
Example: 2,-3,1 represents 2x² - 3x + 1
Example: 1,2,-1 represents x² + 2x - 1
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Sets – Set Operations Visualizer
Sets - Set Operations Visualizer
Enter elements for two sets and perform set operations.
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A
B
Real Numbers-HCF Calculator
Real Numbers - HCF Calculator
Find the Highest Common Factor (HCF) of two numbers using Euclid's algorithm.
Formula Sheet
Formula Sheet
Interactive formula bank for mathematics and science
Algebra Formulas
Quadratic Formula
x = [-b ± √(b² - 4ac)] / 2a
Solution for equations of the form ax² + bx + c = 0
Arithmetic Sequence
aₙ = a₁ + (n - 1)d
nth term of an arithmetic sequence
Geometric Sequence
aₙ = a₁ × rⁿ⁻¹
nth term of a geometric sequence
Sum of Arithmetic Series
Sₙ = n/2 × (a₁ + aₙ)
Sum of first n terms of arithmetic sequence
Sum of Geometric Series
Sₙ = a₁(1 - rⁿ)/(1 - r)
Sum of first n terms of geometric sequence (r ≠ 1)
Exponent Rules
aᵐ × aⁿ = aᵐ⁺ⁿ
Product of powers with same base
Logarithm Properties
logₐ(mn) = logₐ m + logₐ n
Product rule for logarithms
Distance Formula
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Distance between two points in coordinate plane
Geometry Formulas
Pythagorean Theorem
a² + b² = c²
For right triangles, where c is the hypotenuse
Area of Circle
A = πr²
Area of a circle with radius r
Circumference of Circle
C = 2πr
Circumference of a circle with radius r
Area of Triangle
A = (1/2)bh
Area of a triangle with base b and height h
Area of Rectangle
A = l × w
Area of a rectangle with length l and width w
Volume of Cube
V = s³
Volume of a cube with side length s
Volume of Cylinder
V = πr²h
Volume of a cylinder with radius r and height h
Surface Area of Sphere
SA = 4πr²
Surface area of a sphere with radius r
Trigonometry Formulas
Pythagorean Identity
sin²θ + cos²θ = 1
Fundamental trigonometric identity
Sine Rule
a/sinA = b/sinB = c/sinC
For any triangle with sides a, b, c and angles A, B, C
Cosine Rule
a² = b² + c² - 2bc cosA
For any triangle with sides a, b, c and angle A
Tangent Identity
tanθ = sinθ / cosθ
Definition of tangent function
Reciprocal Identities
cscθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ
Reciprocal trigonometric functions
Area of Triangle (Trig)
A = (1/2)ab sinC
Area using two sides and included angle
Angle Sum Identity
sin(A+B) = sinA cosB + cosA sinB
Sine of sum of two angles
Double Angle Formula
sin(2θ) = 2 sinθ cosθ
Double angle formula for sine
Calculus Formulas
Power Rule
d/dx(xⁿ) = nxⁿ⁻¹
Derivative of x raised to a power
Product Rule
d/dx(uv) = u'v + uv'
Derivative of product of two functions
Quotient Rule
d/dx(u/v) = (u'v - uv')/v²
Derivative of quotient of two functions
Chain Rule
d/dx[f(g(x))] = f'(g(x)) × g'(x)
Derivative of composite functions
Derivative of sin x
d/dx(sin x) = cos x
Derivative of sine function
Derivative of cos x
d/dx(cos x) = -sin x
Derivative of cosine function
Power Rule for Integration
∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
Integration of power functions
Fundamental Theorem
∫ₐᵇ f(x) dx = F(b) - F(a)
Where F'(x) = f(x)
Physics Formulas
Newton's Second Law
F = ma
Force equals mass times acceleration
Kinetic Energy
KE = (1/2)mv²
Energy of motion
Potential Energy
PE = mgh
Gravitational potential energy
Ohm's Law
V = IR
Voltage equals current times resistance
Density Formula
ρ = m/V
Density equals mass divided by volume
Work Formula
W = Fd cosθ
Work equals force times displacement times cosine of angle
Power Formula
P = W/t
Power equals work divided by time
Speed Formula
v = d/t
Speed equals distance divided by time
Similar Triangle Construction
Constructing Similar Triangles with Given Ratio
A geometric construction using triangle sides 5cm, 5.4cm, 6cm with 3/4 scale factor
Problem Statement
Construct a triangle with sides 5 cm, 5.4 cm, and 6 cm. Then construct a similar triangle whose sides are \(\frac{3}{4}\) times the sides of the original triangle.
Mathematical Foundation
For similar triangles, the ratio of corresponding sides is constant:
\[
\frac{A'B'}{AB} = \frac{B'C'}{BC} = \frac{A'C'}{AC} = k
\]
where \(k = \frac{3}{4}\) in this case.
The sides of the new triangle will be:
\[
\begin{align*}
5 \times \frac{3}{4} &= 3.75 \text{ cm} \\
5.4 \times \frac{3}{4} &= 4.05 \text{ cm} \\
6 \times \frac{3}{4} &= 4.5 \text{ cm}
\end{align*}
\]
Step-by-Step Construction Process
Part 1: Constructing the Original Triangle
1
Draw the base: Draw segment AB = 6 cm
2
Construct point C using compass: With A as center, radius 5 cm, draw an arc. With B as center, radius 5.4 cm, draw another arc. The intersection point is C.
3
Complete the triangle: Join AC and BC to form triangle ABC.
Part 2: Constructing the Similar Triangle
Method: Dilation from a Point
4
Choose a center of dilation: Select point A as the center.
5
Draw rays: Draw ray from A through B and ray from A through C.
6
Mark points for \(\frac{3}{4}\) ratio: On ray AB, mark point B' such that AB' = \(\frac{3}{4}\) × AB = 4.5 cm. On ray AC, mark point C' such that AC' = \(\frac{3}{4}\) × AC = 3.75 cm.
7
Complete the similar triangle: Join B'C' to form triangle AB'C'.
Verification
Original Triangle Sides
| Side | Length |
|---|---|
| AB | 6.00 cm |
| BC | 5.40 cm |
| AC | 5.00 cm |
Similar Triangle Sides
| Side | Length |
|---|---|
| A'B' | 4.50 cm (6 × 3/4) |
| B'C' | 4.05 cm (5.4 × 3/4) |
| A'C' | 3.75 cm (5 × 3/4) |
Geometric Proof of Similarity
The triangles are similar by Side-Side-Side (SSS) Similarity:
\[
\frac{A'B'}{AB} = \frac{B'C'}{BC} = \frac{A'C'}{AC} = \frac{3}{4}
\]
Interactive Features
In the Geogebra applet, you can:
Adjust the scale factor using the slider
Move vertices of the original triangle
Observe how the similar triangle maintains the ratio
Verify measurements in real-time
Live Geogebra Construction
Interact with the construction below:
Click here to open in a new window
Construction Notes
The Geogebra construction demonstrates:
- Original triangle ABC with sides 5cm, 5.4cm, 6cm
- Similar triangle AB'C' with sides 3.75cm, 4.05cm, 4.5cm
- Dynamic adjustment of the scale factor
- Real-time measurement display