The inverse theorem of the Pythagorean theorem, the inverse theorem of the Pythagorean theorem

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 A + B 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 theorem; Jiang Mingzu in the Three Kingdoms period made a detailed annotation of the Pythagorean theorem in the "Jiang Mingzu Sutra" and gave another proof.

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

We get x +y =c , and a +b =c and a +b =x +y (a).

But a>y, b>x, a +b >x +y (b) a) contradicts (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 , a +b =c =a +b +2ay2ay=0a≠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.

a²+b²=c²

A and B represent right-angled edges, respectively.

c stands for right-angled edge.

If the sum of the squares of the two right-angled edges equals the square of the hypotenuse then.

It can be explained that the triangle ABC is a right triangle.

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 + bThe 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).

(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.

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.

Pythagorean theorem: b 2 = c 2-a 2

Sine theorem: b (sinb) = c (sin90).

In addition to having the properties of a general triangle, it has some special properties:

1. The sum of the squares of the two right-angled sides of a right-angled triangle is equal to the square of the hypotenuse. BAC = 90°, then AB +AC = BC (Pythagorean theorem).

3. In a right-angled triangle, the middle line on the hypotenuse is equal to half of the hypotenuse (that is, the outer center of the right-angled triangle is located at the midpoint of the hypotenuse, and the radius of the circumscribed circle r=c 2). This property is known as the hypotenuse midline theorem of a right triangle.

4. The product of the two right-angled sides of a right-angled triangle is equal to the product of the hypotenuse and the height of the hypotenuse.

Solution: The inverse theorem of the Pythagorean theorem is that in a triangle, if the sum of the squares of the two shorter sides is equal to the square of the longer sides, then the triangle is a right triangle. The angle opposite the longer side is a right angle.