【1st Year Junior High Math】Mastering Plane Geometry!
Hello everyone! Welcome to the world of "Plane Geometry" starting today.
You might be thinking, "Geometry sounds like a hassle because I have to use rulers and compasses..." But in reality, our surroundings are overflowing with "geometric shapes"—from smartphone screens and road signs to the shape of a pizza.
In this chapter, we will learn the basic rules of geometry and the "magic steps" to draw shapes beautifully. Let’s enjoy this together, almost like solving a puzzle!
1. Basics of Lines and Angles
First, let's learn the "vocabulary" and "symbols" for drawing figures. In mathematics, we often represent words with symbols.
(1) Lines, Line Segments, and Rays
・Line \(AB\): A line that passes through points \(A\) and \(B\) and extends infinitely in both directions.
・Line segment \(AB\): A line that connects point \(A\) to point \(B\) with definite endpoints. (This is what we usually call a "line" in everyday life.)
・Ray \(AB\): A line that starts at point \(A\) and extends infinitely in the direction of point \(B\).
(2) Perpendicular and Parallel
・Perpendicular: When two lines intersect at a \(90^{\circ}\) (right) angle. We write this as \(AB \perp CD\).
・Parallel: When two lines never intersect no matter how far you extend them. We write this as \(AB // CD\).
・Distance: The "distance" between a point and a line refers to the length of the "perpendicular line" drawn from the point to the line. Just remember, it’s always the shortest path!
(3) How to Represent Angles
To represent an angle, we write it like \(\angle ABC\). In this case, the middle letter (in this case, \(B\)) is the vertex of the angle. Be careful—this is a common mistake on tests!
【Pro Tip】
When memorizing symbols, visualize them: "\(\perp\)" looks like a right angle, and "//" looks like two lines running side by side. It makes them much harder to forget!
2. Geometric Transformations
Moving a figure to a different location without changing its shape or size is called a "transformation." There are three types.
(1) Translation
Sliding a figure in a fixed direction by a fixed distance. Think of it like "sliding a tray smoothly across a table."
(2) Rotation
Rotating a figure by a certain angle around a specific point (the center of rotation). It’s like "the motion of using a compass to draw a circle."
Specifically, a \(180^{\circ}\) rotation is called a "point-symmetric transformation."
(3) Reflection
Flipping a figure over a single line (the axis of symmetry). Think of it like "seeing yourself in a mirror" or "a stamp impression."
【Common Mistake】
In a reflection, the line segment connecting the original point and the reflected point is perpendicularly bisected by the axis of symmetry. Don't forget these two conditions: "perpendicular" and "equal length"!
3. Basic Construction (Compass Magic)
Drawing figures using only a compass and a straightedge (do not use the ruler markings!) is called "construction." Let’s learn the three "sacred tools" you’ll learn in your first year of junior high.
(1) Perpendicular Bisector
How to draw a line that passes through the exact middle of segment \(AB\) at a right angle.
1. Draw part of a circle from point \(A\).
2. Without changing the radius, draw part of a circle from point \(B\).
3. Connect the two intersection points!
★ Why it’s useful: You can find a point that is equidistant from two points.
(2) Angle Bisector
How to draw a line that cuts an angle exactly in half (bisects it).
1. Draw a circle from the vertex to find the intersection points on the two sides.
2. From those two intersection points, draw circles with the same radius.
3. Connect that intersection to the vertex!
★ Why it’s useful: You can find a point that is equidistant from two sides.
(3) Perpendicular Line
How to draw a line perpendicular to a given line that passes through a specific point. The method is very similar to the angle bisector.
【Fun Fact】
The rule is: do not erase your construction marks (the compass lines). Teachers want to see the "process" of how you drew it!
4. Circles and Sectors
Calculations are finally here! Starting in junior high, we use the symbol \(\pi\) (pi) for the circular constant (\(3.14\dots\)). This is a "magic symbol" that makes calculations much easier!
(1) Circle Formulas (let \(r\) be the radius)
・Circumference \(L\): \(L = 2\pi r\)
・Area of a Circle \(S\): \(S = \pi r^2\)
How to remember: "Circumference is \(2\pi r\) (two-pi-r), area is \(\pi r^2\) (pi-r-squared)." Try to memorize it with a rhythm!
(2) Sectors
A sector is a "cut piece of a circle." Imagine a slice of pizza.
The angle at the center is called the central angle (\(a^{\circ}\)).
・Arc length \(\ell\): Circumference \(\times \frac{a}{360}\)
・Area \(S\): Area of a circle \(\times \frac{a}{360}\)
In other words, just think: "What fraction of the whole circle is this?"
【Pro Technique】
There is another formula for the area of a sector:
\(S = \frac{1}{2} \ell r\) (where \(\ell\) is the arc length and \(r\) is the radius)
If you know this, you can calculate the area even if you don't know the central angle, which saves time on tests!
5. Final Thoughts: Tips for Solving Plane Geometry
It might feel difficult at first, but "getting your hands moving" is the best way to improve in geometry.
・Write the information from the problem directly onto your diagram (like marking \(//\) or \(\perp\)).
・Don't be stingy with your compass; use it to draw clear, large marks.
・If you forget a formula, try drawing the whole circle and asking, "What fraction is this?"
If you master this chapter, the "proofs" and "solid geometry" you'll learn in the 2nd and 3rd years will be much easier. Start by giving that compass a spin—I’m rooting for you!