"The Converse Theorem of Pythagorean Theorem" PPT courseware of Pythagorean Theorem

"The Converse Theorem of Pythagorean Theorem" PPT courseware of Pythagorean Theorem

Review the past and learn the new

Pythagorean Theorem: If the two right-angled sides of a right triangle are a and b, and the length of the hypotenuse is c, then a2+b2=c2.

think:

Conversely, if the lengths of the three sides a, b, and c of a triangle satisfy a2+b2=c2, then what is the shape of this triangle?

The converse of the Pythagorean theorem

If the side lengths a, b, and c of a triangle satisfy a2+b2=c2, then the triangle is a right triangle.

Reciprocal proposition:

Among two propositions, if the hypothesis of the first proposition is the conclusion of the second proposition, and the conclusion of the first proposition is the hypothesis of the second proposition, then these two propositions are called reciprocal propositions.

If one of them is called the original proposition, then the other is called its converse proposition.

Reciprocity theorem:

If the converse of a theorem is proven to be true, then it is also a theorem. These two theorems are called reciprocal theorems, and one of them is called the converse of the other.

give it a try

State the converse of the following proposition. Are the converses of these propositions true?

(1) Two straight lines are parallel and their internal offset angles are equal.

Converse proposition: Internal angles are equal and two straight lines are parallel. Established

(2) If two real numbers are equal, then their squares are equal.

Converse proposition: If the squares of two real numbers are equal, then the two real numbers are equal. Not true

(3) If two real numbers are equal, then their absolute values ​​are equal.

Converse proposition: If the absolute values ​​of two real numbers are equal, then the two real numbers are equal. Not true

(4) Corresponding angles of congruent triangles are equal.

Converse proposition: Two triangles with equal corresponding angles are congruent triangles. Not true

Example analysis

Example 1 Determine whether the triangle composed of a, b, c is a right triangle:

(1) a＝15, b＝8, c＝17

(2) a＝13, b＝15, c＝14

Analysis: Based on the converse theorem of the Pythagorean theorem, to determine whether a triangle is a right triangle, just look at whether the sum of the squares of the two smaller sides is equal to the square of the largest side.

Solution: ∵152+82=225+64=289

172＝289

∴ 152+82＝172

∴This triangle is a right triangle

Class exercises

Determine whether the triangle composed of line segments a, b, and c is a right triangle:

(1) a=15, b=8, c=17;

(2) a=m2-n2, b=m2+n2, c=2mn (m>n, m and n are positive integers)

Solution; (1)∵a2 = 225,

b2 = 64, c2 = 289

Also ∵ 225 + 64 = 289

∴ a2 + b2 = c2

That is: the triangle is a right triangle

(2)∵a2 = (m2 - n2 )2 = m4 - 2m2n2 + n4, b2 = (m2 + n2 )2 = m4 + 2m2n2 + n4,

c2 = (2mn )2 = 4m2n2

Also ∵m4 - 2m2n2 + n4 + 4m2n2 = m4 + 2m2n2 + n4

∴ a2 + c2 = b2

That is: the triangle is a right triangle

Practice in class:

1. Connect the three wooden rods of the following lengths end to end to form a right triangle ( )

(A)1, 2, 3 (B)4, 6, 8 (C)5, 5, 4 (D)15,12, 9

2. If line segments a, b, and c can form a right triangle, their ratio may be ( )

(A) 3:4:7; (B) 5:12:13;

(C) 1:2:4; (D) 1:3:5.

3. The three sides of the triangle are a, b, and c respectively, and satisfy (a+b)2-c2=2ab, then the triangle is: ( )

A. A right triangle; B. An acute triangle;

C. is an obtuse triangle; D. is an isosceles right triangle.

The three positive integers satisfying a2+b2=c2 are called Pythagorean numbers.

Can you write down the commonly used Pythagorean numbers?

3, 4, 5; 5, 12, 13;

6, 8, 10; 7, 24, 25;

8, 15, 17; 9, 40, 41

How to determine if a triangle is a right triangle

Angles: A triangle with one right angle is a right triangle.

Sides: If the lengths of the three sides of a triangle are a, b, and c such that a2+b2=c2, then the triangle is a right triangle.

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