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Mathematics EN 9th grade: What is a geometric locus?

What is a geometric locus?

A geometric locus is the set of all points that satisfy a given condition.

In simple terms:

It is the set of points that obey a specific rule.

Example: The points that are at a fixed distance from a point form a circle.

1. Geometric loci in the plane

Main idea: In the plane, a geometric locus can be:

A line

A circle

A region of the plane

Examples: Points at a fixed distance from a point → circle

Points equidistant from two points → perpendicular bisector

2. Perpendicular Bisector

- Definition

The perpendicular bisector of a segment is the line that:

Passes through the midpoint of the segment

Is perpendicular to the segment

📌 Important property:

All points on the perpendicular bisector are equidistant from the endpoints of the segment.

👉 If a point belongs to the perpendicular bisector, then it is equidistant from both endpoints.

📌 Example

Given the segment AB:

The perpendicular bisector passes through the midpoint of AB

For any point P on the perpendicular bisector, it is true that:

PA = PB

3. Angle Bisector of a Convex Angle

- Definition

The angle bisector is the ray that divides the angle into two equal angles.

📌 Important Property:

All points on the bisector are equidistant from the sides of the angle.

📌 Example: If an angle measures 80°, the bisector divides it into:

40° + 40°

If a point P belongs to the bisector:

👉 It is equidistant from both sides of the angle.

QUICK SUMMARY

Geometric locus: set of points that satisfy a condition. Perpendicular bisector:

Line perpendicular to the segment.

Passes through the midpoint.

Points verify PA = PB

Angle bisector:

Divides the angle into two equal parts

Points are equidistant from the sides of the angle

Did you know?

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Explore more about What are rational numbers?
They are numbers that can be written in fraction form:
$ \frac{a}{b}, \quad b \neq 0 $
Examples:
$ \frac{3}{4}, -2, 0{,}5, -\frac{7}{3} $
valor absoluto
The valor absoluto represents the distance to zero:
$ |a| = a $ se $ a \geq 0 $
$ |a| = -a $ se $ a
Examples:
\( |5| = 5 $
$ |-5| = 5 $
Simétrico
The Simétrico of a number is:
$ -a $
Examples:
$ 3 \rightarrow -3 $
$ -7 \rightarrow 7 $
Adição e subtração
Addition with the same sign
Add the values and keep the sign
Examples:
$ 3 + 5 = 8 $
$ -4 + (-6) = -10 $
Soma com sinais contrários
Subtract the values and take the sign of the number with the greatest absolute valueNL#
Examples:
$ 7 + (-3) = 4 $
$ -8 + 5 = -3 $
Soma de números simétricos
The result is always zero
$ a + (-a) = 0 $
Examples:
$ 5 + (-5) = 0 $
Subtração (diferença)
It is transformed into the sum of the opposite
$ a - b = a + (-b) $
Examples:
$ 7 - 3 = 7 + (-3) = 4 $
$ 5 - (-2) = 5 + 2 = 7 $
Divisão de números racionais
Main rule
Multiply by the reciprocal
$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} $
Inverso de um número racional
Swap the numerator and denominator
$ \text{Inverso de } \frac{a}{b} = \frac{b}{a}, \quad a \neq 0 $
Examples:
$ \frac{3}{4} \rightarrow \frac{4}{3} $
$ 2 = \frac{2}{1} \rightarrow \frac{1}{2} $
Important:
$ 0 \text{ não tem inverso} $
Examples de divisão
$ \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} $
TIPS
✔ Same signs → add and keep the sign
✔ Different signs → subtract and take the sign of the number with the greatest absolute value
✔ Subtraction → convert to addition
✔ Division → multiply by the reciprocal
✔ Número + simétrico: $ a + (-a) = 0 $
✔ Valor absoluto: $ |a| \geq 0 $
..

What are rational numbers?
They are numbers that can be written in fraction form:
$ \frac{a}{b}, \quad b \neq 0 $
Examples:
$ \frac{3}{4}, -2, 0{,}5, -\frac{7}{3} $
valor absoluto
The valor absoluto represents the distance to zero:
$ |a| = a $ se $ a \geq 0 $
$ |a| = -a $ se $ a < 0 $
Examples:
$ |5| = 5 $
$ |-5| = 5 $
Simétrico
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Pronome possessivo
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Pronome indefinido
Indicates vague quantity or identity: alguém, ninguém, tudo, algum, certo.
Pronome relativo
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Word that replaces or accompanies a noun, avoiding repetition and indicating person, possession, quantity, or relation.
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Subclasses de pronomes
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