"The Converse Theorem of the Pythagorean Theorem" PPT courseware 3

"The Converse Theorem of the Pythagorean Theorem" PPT courseware 3

1. Knowledge connection:

Question 1. Can you tell what characteristics a right triangle has?

(1) One angle is a right angle:

(2) The right-angled side of 30 degrees is equal to half of the hypotenuse;

(3) Pythagorean Theorem: The sum of the squares of the two right angle sides is equal to the square of the hypotenuse.

(4) Two acute angles are complementary to each other;

2. Question: What conditions must be met for a triangle to be a right triangle?

(1) From the perspective of angles: a triangle with one right angle is a right triangle;

(2) We have studied the Pythagorean Theorem. We know that the three sides of a right triangle have a certain quantitative relationship. Can we use the relationship between the three sides of a triangle to determine whether it is a right triangle without using the angles?

2. Preliminary exploration of new knowledge:

Activity 1: The following three sets of data are the three sides a, b, and c of a triangle.

(1)3cm, 4cm, 5cm;

(2)6cm,8cm,10cm;

(3)5cm, 12cm, 13cm.

Question: (1) Do these three sets of numbers satisfy a²+b²=c²?

(2) Taking the first two sides of each group of numbers as

Construct a right triangle with right angled sides and try to calculate the hypotenuse

Proposition 2

If the lengths of the three sides a, b, and c of the triangle satisfy a2 + b2 = c2

Then this triangle is a right triangle.

Proposition 1

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

It is known that in △ABC, AB=c BC=a CA=b and a²+b²=c²

Prove: △ ABC is a right triangle

Proof: Draw a △A’B’C’ so that ∠ C’=90°, B’C’=a, C’A’=b

In △ ABC and △ A’B’C’

BC=a=B’C’

CA=b=C’A’

AB=c=A’B’

∴ △ ABC ≌△ A’B’C’ (SSS)

∴ ∠ C= ∠ C’ (corresponding angles of congruent triangles are equal)

∴ ∠C= 90°

∴ △ ABC is a right triangle (definition of right triangle)

Theorem and Converse Theorem

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

We have already learned some reciprocal theorems, such as:

Pythagorean theorem and its converse,

If two straight lines are parallel, the internal offset angles are equal; if the internal offset angles are equal, the two straight lines are parallel.

Think about it:

What is the relationship between the reciprocity proposition and the reciprocity theorem?

judge:

Is the triangle composed of line segments t, m, n a right triangle?

(1)t=15 m=8 n=17;

(2)t=10 m=8 n=16;

(3) t=13 m=4 n=15.

Comments:

From a²+b²=c² we know that c>a, and c>b.

method:

Just check whether the sum of the squares of the two smaller side lengths is equal to the square of the largest side length.

Is the triangle below with sides a, b, and c a right triangle? If so then which angle is a right angle?

(1) a=25 b=20 c=15 ____ _____ ;

(2) a=13 b=14 c=15 ____ _____ ;

(3) a=1 b=2 c=√3 ____ _____;

(4) a:b: c=3:4:5 _____ _____ ;

Like 25, 20, and 15, three positive integers that can become the lengths of the three sides of a right triangle are called Pythagorean numbers.

practise

3． If the three sides of △ABC are a, b, and c respectively and satisfy

a2＋b2＋c2+50＝6a＋8b+10c,

Determine the shape of △ABC.

This triangle is a right triangle.

Variation training

Activity 7

(1). As shown in the figure: AD⊥CD, AC⊥BC, AB=13, CD=3, AD=4.

Find: (1) Find the length of AC (2) Find the length of BC

(2). As shown in the figure, AD⊥CD, AB=13, BC=12, CD=3, AD=4.

Find: (1) Find the length of AC (2)∠ACB.

6. Learn and gain:

1. What new insights have you gained through studying this lesson?

2. What is the relationship between the theorem learned in this lesson and the Pythagorean Theorem learned previously?

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