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Geometry is the study of shapes, sizes, positions, and dimensions of things. It helps us understand spatial relationships and has been crucial in fields such as engineering, art, and architecture.
A point defines a location in space, a line is an infinite series of points extending in both directions, and a plane is a flat surface that extends indefinitely in two dimensions.
An angle is formed when two rays share a common endpoint. Angles are an important part of geometry and can be classified as:
Acute Angle: An angle of 45°.
Right Angle: An angle of 90°.
Obtuse Angle: An angle of 120°.
Straight Angle: An angle of 180°.
Reflex Angle: An angle of 240°.
A triangle is a polygon with three edges and three vertices. Types of triangles include:
Equilateral Triangle: All sides and angles are equal (60° each).
Isosceles Triangle: Two sides are of equal length.
Scalene Triangle: All sides are of different lengths.
Right Triangle: One of the angles is exactly 90°.
A polygon is a 2D shape with straight sides. Common polygons include:
Quadrilateral: A four-sided polygon, like a square.
Pentagon: A five-sided polygon.
Hexagon: A six-sided polygon.
Octagon: An eight-sided polygon.
Types include equilateral (all sides equal), isosceles (two sides equal), and scalene (all sides unequal). The sum of the angles in a triangle is always 180°.
Key properties include:
Includes the distance formula d = √((x₂ - x₁)² + (y₂ - y₁)²) and midpoint formula M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
1. What is the missing angle in a triangle where two angles measure 50° and 60°?
2. What is the radius of a circle with a diameter of 18 cm?
3. In a cyclic quadrilateral, if one angle is 110°, what is the measure of its opposite angle?
4. In the diagram below, if angle A at the circumference is 40°, what is angle B at the center?
5. In a coordinate plane, what is the midpoint of points (2, 3) and (6, 7)?
1. Find the distance between points A(2, 3) and B(5, 7).
2. What is the midpoint of points A(-2, 4) and B(6, -4)?
3. Determine the slope of the line passing through points P(3, -2) and Q(7, 6).
Used to find the distance between two points on a Cartesian plane.
Finds the exact middle point between two points.
Determines the steepness of a line. Positive slope = upward direction, negative slope = downward.
Expresses a linear relationship. The slope \( m \) and y-intercept \( c \) determine the line's orientation.
This describes a circle with center \( (h, k) \) and radius \( r \).
Problem: Calculate the distance between points A(3, 4) and B(7, 1).
Solution:
Using the distance formula, substitute the points:
d = √((7 - 3)^2 + (1 - 4)^2)
= √(4^2 + (-3)^2)
= √(16 + 9)
= √25
= 5
Answer: The distance between points A and B is 5 units.
Problem: Find the midpoint between points C(-5, 2) and D(3, 8).
Solution:
Using the midpoint formula:
M = ( (x1 + x2)/2 , (y1 + y2)/2 )
= ( (-5 + 3)/2 , (2 + 8)/2 )
= (-2/2 , 10/2)
= (-1, 5)
Answer: The midpoint is (-1, 5).
1. Find the distance between points (2, 1) and (6, 5).
2. What is the slope of the line passing through points (3, 2) and (7, 10)?
1. Find the distance between the points (3, 4) and (8, -2).
2. Determine the midpoint of the line segment joining points (-1, 2) and (5, -6).
3. What is the slope of the line that passes through the points (7, -1) and (3, 5)?
4. Find the equation of the line with a slope of 3 and passing through the point (2, -5).
5. A circle has a center at (4, -2) and passes through the point (7, 2). What is the radius of the circle?
6. Which of the following is the equation of a circle centered at (2, 3) with radius 4?
Polygons are shapes with straight sides that are closed and can be represented using vertices in the Cartesian plane. Each vertex is a coordinate pair (x, y).
To define a polygon, list its vertices in order, either clockwise or counterclockwise.
Enter coordinates to see your polygon plotted:
Test your understanding with these questions:
The Cartesian plane is a two-dimensional plane formed by the intersection of a vertical line (the y-axis) and a horizontal line (the x-axis). These lines intersect at a point called the origin, which has coordinates (0, 0).
Polygons can be plotted on the Cartesian plane using their vertices, which are defined by pairs of coordinates (x, y). This allows for the calculation of properties like side lengths, perimeters, areas, and centroids using mathematical formulas.
Distance = √((x2 - x1)² + (y2 - y1)²)
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Area = ½ |Σ (xᵢyᵢ₊₁ - yᵢxᵢ₊₁)|
Consider a triangle with vertices A(2, 3), B(5, 7), and C(1, 4). Using the distance formula, calculate the side lengths:
√((5 - 2)² + (7 - 3)²) = √(9 + 16) = 5√((5 - 1)² + (7 - 4)²) = √(16 + 9) = 5√((2 - 1)² + (3 - 4)²) = √(1 + 1) = √2Try plotting your own polygon by entering the coordinates of its vertices:
Side Lengths:
Perimeter:
Area:
The Cartesian plane is a tool for plotting points and shapes. It consists of an x-axis (horizontal) and a y-axis (vertical), intersecting at the origin (0, 0). You can define polygons on the plane using their vertices and explore properties such as side lengths, perimeter, and area.
Enter the vertices of your polygon to see it plotted on the Cartesian plane. Ensure the coordinates form a concave polygon.
Type: Concave
Side Lengths:
Perimeter:
Area:
A hyperbola is a type of conic section formed by the intersection of a double cone with a plane. Unlike ellipses or circles, hyperbolas consist of two disjoint curves, known as branches. Each branch is a mirror image of the other.
The hyperbola's equation depends on its orientation:
Plot a hyperbola by entering the parameters below:
Pythagoras' Theorem applies to right-angled triangles and states: The square of the hypotenuse is equal to the sum of the squares of the other two sides.
Real-world applications include calculating distances, designing ramps, and solving engineering problems.
Explore angles formed by intersecting lines dynamically.
Use the calculator below to practice finding terms and sums in arithmetic and geometric sequences.
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (denoted as d).
The general term of an arithmetic sequence is given by:
an = a1 + (n - 1)d
an: the nth terma1: the first termd: the common differencen: the term number
Consider the sequence: 2, 5, 8, 11, 14...
Here, the common difference d = 3, and the first term a1 = 2.
Using the formula: an = 2 + (n - 1) × 3, you can calculate any term in the sequence.
A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a constant. This constant is called the common ratio (denoted as r).
The general term of a geometric sequence is given by:
an = a1 × rn-1
an: the nth terma1: the first termr: the common ration: the term number
Consider the sequence: 3, 6, 12, 24, 48...
Here, the common ratio r = 2, and the first term a1 = 3.
Using the formula: an = 3 × 2n-1, you can calculate any term in the sequence.
Test your knowledge of advanced geometry concepts! Solve each question, then click the "Show Solution" button to see the formula and detailed calculation.
Question 1: Find the area of a triangle with base \( b = 10 \) cm and height \( h = 8 \) cm.
Question 2: Calculate the volume of a sphere with radius \( r = 5 \) cm.
Question 3: Determine the length of the hypotenuse of a right triangle with legs \( a = 6 \) cm and \( b = 8 \) cm.
Question 4: Find the area of a circle with radius \( r = 7 \) cm.
Question 5: Calculate the volume of a cylinder with radius \( r = 3 \) cm and height \( h = 10 \) cm.
Question 6: Find the total surface area of a cube with side length \( a = 4 \) cm.
Question 7: Calculate the diagonal of a square with side length \( a = 12 \) cm.
Question 8: Determine the perimeter of a rectangle with length \( l = 15 \) cm and width \( w = 10 \) cm.
Question 9: Find the lateral surface area of a cone with radius \( r = 5 \) cm and slant height \( l = 12 \) cm.
Question 10: Calculate the area of a parallelogram with base \( b = 14 \) cm and height \( h = 9 \) cm.
The midpoint formula is used to find the midpoint between two points:
\( M(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
This formula helps in determining the exact center between two points on a Cartesian plane, which is crucial for tasks like dividing a line segment into equal parts.
The distance formula calculates the distance between two points:
\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
This formula is derived from the Pythagorean theorem and is used to measure the shortest path between two points on a plane.
The slope formula helps to find the steepness or incline of a line between two points:
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Knowing the slope is important for understanding the direction and rate of change between points.