【Grade 9】 Mathematics: The Pythagorean Theorem
Hello! We’re reaching the home stretch of Grade 9 mathematics. The topic we’re covering today, the "Pythagorean Theorem," is one of the most famous theorems in the history of math.
Many people feel like, "I'm just not good at geometry..." but don't worry! The foundation of this theorem is very simple. You only need to understand "the relationship between the lengths of the three sides of a right triangle." Once you grasp this, you’ll be able to use calculation to determine things like the height of tall buildings that you can't measure directly, or distances on a map!
1. What is the Pythagorean Theorem?
The Pythagorean Theorem is a special rule that can only be used with right triangles.
If the lengths of the two sides that form the right angle are \(a\) and \(b\), and the length of the longest side (called the hypotenuse) is \(c\), the following relationship holds:
【Formula】 \(a^2 + b^2 = c^2\)
In other words, "the square of the two shorter sides added together equals the square of the longest side!" Simple, right?
Important Terms
- Right Triangle: A triangle that has one 90-degree angle.
- Hypotenuse: The longest side, located directly opposite the right angle.
Pro Tip: When using the formula, the trick is to first identify which side is the "hypotenuse (c)"! The hypotenuse is always the side opposite the right angle.
【Fun Fact】 Pythagoras' Discovery
It is said that this was discovered by the Greek mathematician Pythagoras about 2,500 years ago. In ancient times, people used this theorem by stretching ropes on the ground to create perfect right angles for building structures.
2. Let’s try calculating side lengths
Now, let’s practice using the formula with some calculations.
Example 1: Finding the hypotenuse
Find the length of the hypotenuse \(x\) for a right triangle where the two sides forming the right angle are \(3cm\) and \(4cm\).
Steps to solve:
- Substitute the numbers into the formula \(a^2 + b^2 = c^2\).
\(3^2 + 4^2 = x^2\) - Calculate.
\(9 + 16 = x^2\)
\(25 = x^2\) - Find the number that equals 25 when squared (since \(x > 0\)).
\(x = 5\)
Answer: \(5cm\)
Example 2: Finding a side other than the hypotenuse
Find the length of the remaining side \(x\) for a right triangle where the hypotenuse is \(13cm\) and another side is \(12cm\).
Steps to solve:
- Substitute into the formula (since the hypotenuse is \(13\), the right side of the equation becomes \(13^2\)).
\(x^2 + 12^2 = 13^2\) - Calculate.
\(x^2 + 144 = 169\) - Subtract to find \(x^2\).
\(x^2 = 169 - 144\)
\(x^2 = 25\) - Find the number that equals 25 when squared.
\(x = 5\)
Answer: \(5cm\)
Advice: Memorizing common squares (like \(11^2=121, 12^2=144, 13^2=169, 15^2=225\)) will significantly speed up your calculations!
3. The Converse of the Pythagorean Theorem
This is the rule that states: "If the lengths of the three sides satisfy \(a^2 + b^2 = c^2\), then the triangle is a right triangle."
For example, if you have a triangle with side lengths of \(6cm, 8cm, 10cm\):
\(6^2 + 8^2 = 36 + 64 = 100\)
\(10^2 = 100\)
Since both sides equal \(100\), you know that this triangle is "a right triangle with a hypotenuse of 10cm!"
4. Test Favorites! "Special Right Triangles"
In junior high school math, two types of right triangles that match the shapes of set squares (triangular rulers) appear very frequently. If you memorize these ratios, you can find the answers without even needing to calculate!
① The 45°-45°-90° Isosceles Right Triangle
The ratio of the sides is \(1 : 1 : \sqrt{2}\).
(Example: If the shorter side is \(5cm\), the hypotenuse is \(5\sqrt{2}cm\))
② The 30°-60°-90° Right Triangle
The ratio of the sides is \(1 : \sqrt{3} : 2\).
Note: The longest side (hypotenuse) is \(2\)! Since \(\sqrt{3}\) is about \(1.73\), it is shorter than \(2\). Be careful not to mix them up.
【Tips for Memorization】
・For \(1 : 1 : \sqrt{2}\), just remember the ratio pattern.
・For \(1 : \sqrt{3} : 2\), remember "1, 2, 3" and remember to put the root on the 3!
5. Common Mistakes (Check these!)
It might feel difficult at first, but you'll be fine if you watch out for these two things!
- Mistaking the hypotenuse: Always place the longest "hypotenuse" into the \(c\) position (the right side of the equals sign) in the formula.
- Forgetting the root (square root): If you get \(x^2 = 7\), the answer is not \(7\), it is \(x = \sqrt{7}\). Don't forget to "add the root" at the end!
- Addition vs. Subtraction errors: You use addition when finding the hypotenuse and subtraction when finding a side that is not the hypotenuse. Look at the diagram carefully when you think about the problem.
Summary: Today’s Key Points
1. When you see a right triangle, think of \(a^2 + b^2 = c^2\)!
2. Be sure to correctly identify the longest side (the hypotenuse)!
3. Memorize the special ratios \((1 : 1 : \sqrt{2})\) and \((1 : \sqrt{3} : 2)\)!
The Pythagorean Theorem will also be very useful for future topics like "spatial figures" (calculating diagonals of 3D objects, etc.). Let’s master it with flat triangles first. I’m rooting for you!