What is the inverse application of the Pythagorean theorem?

If the three sides of a triangle are related to the lengths a, b, and c, then the triangle is a right triangle." This proposition is called the inverse theorem of the Pythagorean theorem.

A thorough, complete and accurate understanding of the above theorems can be obtained from the following aspects:

1. From the expression, it can be seen that it is a right triangle with c as the hypotenuse (i.e., a right triangle. If the expression is changed, it is a right triangle with a hypotenuse (or. Otherwise, it will lead to errors, for example, if a right triangle can be formed, some students may compare the sum of the squares of the two sides with the square of the third side, so they think that they cannot form a right triangle.

However, in this problem, it is easy to see that the b side is the largest, so we can only find whether it is equal to or not. In fact, it can form a right triangle with b as the hypotenuse.

Exercise: The three sides are to determine whether it is a right triangle.

2. If a, b, and c are satisfied, then ma, mb, and mc(m>0) are sides that also form a right triangle.

From this it is necessary to keep in mind some common Pythagorean numbers:

3，4，5；6，8，10；…, 3n, 4n, 5n (n is a positive integer).

5，12，13；10，24，26；…, 5n, 12n, 13n (n is a positive integer).

7，24，25；……7n, 24n, 25n (n is a positive integer).

8，15，17；……8n, 15, 17n (n is a positive integer).

9，40，41；……9n, 40n, 41n (n is a positive integer).

Exercise: The three sides are 15, 20, 25 when they are judged to be shaped.

3. The Pythagorean theorem is often used to determine the right triangle by the inverse theorem, which sometimes avoids the complicated polynomial deformation. For example, the three sides are , and it is verified that it is a right triangle.

Analysis: It is more convenient to use the squared difference formula to calculate the side length, so the calculated values are then compared. Namely.

Whereas. Therefore, it is a right triangle.

Fourth, the inverse of the Pythagorean theorem transforms the features of the numbers in the triangle into the features of the graph (one of which is a right angle) through algebraic operation, so that the properties of the right triangle can be further used to solve related geometric problems.

Example 1 In , d is the midpoint of AB, ac=12, bc=5, cd=, and is verified to be a right triangle.

Analysis: From the given data, 5, 12, and 13 are a set of Pythagorean numbers, and for this reason, the midline cd is doubled, i.e., the extension of cd to c "makes cc" = 2cd = 13.

Easy proof: so bc"=ac=12, in there.

So. So.

Therefore, it is a right-angled triangle.

Example 2 Known: in, , point p and pa=3, pb=1, pc=2, verify:.

Analysis: Rotate 90° counterclockwise around point C to the position, even pp", in RT, so. At. Composed.

Old. Again, therefore.

The sum of the squares of the two sides of the triangle is equal to the square of the third side.

Just use the sum of the squares of the two sides of the triangle to be equal to the square of the third side to prove that the triangle is a right triangle.

I learned very little, hehe.

The sum of the squares of the two sides is equal to the square of the third side, and the triangle formed by the three sides is a right triangle, and the long side is an hypotenuse.

The inverse theorem of the Pythagorean theorem is that if the sum of the squares of two right-angled sides of a triangle is equal to the square of the hypotenuse, then the triangle is a right-angled triangle.

The inverse theorem of the Pythagorean theorem is a simple way to determine whether a triangle is obtuse, acute, or right, where ab=c is the longest side.

If a + b = c then abc is a right triangle.

If a +b > c then abc is an acute triangle (if ab=c is the longest side without the previous condition, then the formula only satisfies c is an acute angle).

If a + b <The specific explanation of the Pythagorean theorem is as follows:

1. The Pythagorean theorem, also known as the quotient theorem, Pythagorean theorem, Pythagorean theorem, and Pythagorean theorem, is a basic and important theorem in plane geometry. The Pythagorean theorem is one of the important mathematical theorems discovered and proven by mankind in the early days.

2. The Pythagorean theorem states that the sum of the squares of the lengths of the two right-angled sides of a right-angled triangle on a plane (known as hook length and strand length) is equal to the square of the hypotenuse length (known as chord length in ancient times).

3. Conversely, if the sum of the squares of the two sides of a triangle on a plane is equal to the square of the length of the third side, it is a right triangle (the side opposite the right angle is the third side).

If the sum of the squares of the two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle.

The angle on the longest side is a right angle. The inverse theorem of the Pythagorean theorem is a simple way to tell if a triangle is acute, right, or obtuse.

If c is the longest side and a +b = c then abc is a right triangle. If A+B >C, then ABC is an acute triangle. If the Pythagorean theorem is a basic geometric theorem, in China, the formula and proof of the Pythagorean theorem are recorded in the Zhou Sutra, which is said to have been discovered by Shang Gao in the Shang Dynasty, so it is also called Shang Gao's theoremJiang Mingzu in the Three Kingdoms period made a detailed annotation of the Pythagorean theorem in the "Jiang Mingzu Sutra" and gave another proof.

Pythagorean theorem: In a right-angled triangle on a plane, the square of the length of the two right-angled sides adds up to the square of the hypotenuse length.

As shown in the figure below, i.e., a + b = c ).

Example: For example, in the right triangle of the figure above, the side length of a is 3 and the side length of b is 4, then we can use the Pythagorean theorem to calculate the side length of c.

From the Pythagorean theorem, a + b = c 3 +4 = c

i.e., 9 + 16 = 25 = c

c = 25 = 5

So we can use the Pythagorean theorem to calculate that the side length of c is 5.

The inverse theorem of the Pythagorean theorem:

The inverse theorem of the Pythagorean theorem is a simple way to determine whether a triangle is obtuse, acute, or right, where ab=c is the longest side.

If a + b = c then abc is a right triangle.

If a +b > c then abc is an acute triangle (if ab=c is the longest side without the previous condition, then the formula only satisfies c is an acute angle).

If a + b <

Inverse theorem proof of the Pythagorean theoremThe inverse theorem of the Pythagorean theorem is a simple way to tell if a triangle is acute, right, or obtuse. If c is the longest side and a +b = c then abc is a right triangle. If A+B >C, then ABC is an acute triangle. If a + b <

According to the cosine theorem, in ABC, cosc=(a +b -c) 2AB.

Since a + b = c , cosc = 0;

Because 0°< c<180°, c=90°. (Proof complete) is known in ABC, verify C=90°

Proof: AH BC in H

If c is an acute angle, let bh=y, ah=x

x +y =c , a +b =c , a +b =x +y (a).

But a>y, b>x, a +b >x +y (b) (a) contradict (b), c is not an acute angle.

If c is an obtuse angle, let hc=y, ah=x

A +b =c =x +(a+y) =x +y +2ay+a x +y =b and a +b =c =a +b +2ay2ay=0 a≠0, y=0

This contradicts that c is an obtuse angle, and c is not an obtuse angle.

To sum up, c must be a right angle.

The Pythagorean theorem is a basic theorem in mathematics that we learn and one of the important theorems to solve plane geometry problems. It is expressed as follows: In a right triangle, the square of the right sides is equal to the sum of the squares of the other two sides.

However, anyone with a certain mathematical foundation knows that this is only one of the expressions of the Pythagorean theorem, and it has a range of different formulations, and further, the inverse theorem of the Pythagorean theorem. So, what is the inverse theorem of the Pythagorean theorem? The inverse theorem of the Pythagorean theorem states that if the side lengths of the three sides of a triangle meet the conditions of the Pythagorean theorem, then the triangle must be a right triangle.

In simple terms, the inverse theorem is the reverse of the Pythagorean theorem. If the length of the sides in a triangle conforms to the formula a +b =c, then it turns out that the triangle must be a right triangle.

So, how is the inverse theorem of the Pythagorean theorem derived? The earliest method of proof was based on the method of counterproof. Suppose that the side lengths of the three sides of a triangle meet the conditions of the Pythagorean theorem, but the triangle is not a right triangle, then a contradiction is obtained.

Because the Pythagorean theorem only applies to right triangles, if the triangle is not a right triangle, then the Pythagorean theorem does not hold. Therefore, this assumption is wrong, and this triangle must be a right triangle.

In addition to the method of refutation, there is also a common method of proof in the world, using trigonometric functions to prove. According to the sine theorem and the cosine theorem, the ratio of the square root of the cosine of the inner angle of a triangle equal to the sum of the squares of the corresponding side lengths can be obtained. If the cosine values of the three interior angles correspond to the ratio of the square roots of the sum of the squares of the three side lengths to the Pythagorean theorem, then the triangle is a right triangle.

This method of proof requires a certain amount of mathematical knowledge and skill, but it has a wider scope of application than the counterproof method.

In conclusion, the inverse theorem of the Pythagorean theorem is a fundamental mathematical theorem that states that a triangle can only be a right triangle if the side length meets the conditions of the Pythagorean theorem. Understanding and mastering it can help us better solve plane geometry problems, and it is also an important foundation for us to learn mathematics.

(m 2-n 2) 2+(2mn) 2=m 4-2m 2n 2+n 4+4m 2n 2=m 4+2m 2n 2+n 4=(m 2-n 2) 2 satisfies the inverse theorem of the Pythagorean theorem.