"Can you get a right triangle?" Pythagorean Theorem PPT Courseware 2

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"Can you get a right triangle?" Pythagorean Theorem PPT Courseware 2

The ancient Egyptians used the following method to obtain right angles:

They used 13 equally spaced knots to divide a rope into 12 equal-length sections. A craftsman held the 1st and 13th knots of the rope at the same time, and two assistants held the 4th and 8th knots respectively. Knot and tighten the rope, you will get a right triangle, the right angle is at the 4th knot.

Draw a picture:

Use each of the following sets of numbers to make a triangle with three sides (unit: cm)

(1)3,4,5 (2)3,4,6 (3)4,5,6 (4)5,12,13

Find it:

Do these four sets of numbers satisfy a²+b²=c²?

Measure:

Use a protractor to determine the shape of the triangle you drew.

Take a guess:

Let us guess what relationship the lengths of the three sides of a triangle should satisfy before this triangle can be a right triangle?

See who can figure it out

Think of any three numbers and ask: the sum of the squares of two of the numbers is equal to the square of the third number.

Hands-on drawing: The three numbers you came up with in the above question are the side lengths, and draw a triangle.

The two shorter sides in the above question are right-angled sides, so draw a right-angled triangle.

Cut out the two triangles you drew above and put them together. What did you find?

Can the following sets of numbers be used as the three sides of a right triangle? Tell us your reasons.

(1) 9,12,15 (2)15,36,39

(3)12,35,36 (4)12,18,22

We know that the length ab of the two right-angled sides of a right triangle and the length c of the hypotenuse satisfy the equation: a²+b²=c², and we can find some positive integer arrays (ie, Pythagorean arrays) that satisfy this equation. So how many Pythagorean arrays are there? Do they follow a certain pattern? In fact, there are countless Pythagorean arrays. The following is a way to find the Pythagorean array: for any two positive integers m,n (m>n), the three numbers m²+n², m²-n² and 2mn are a set of Pythagorean arrays. Can you verify this conclusion?

The 17th-century French mathematician Fermat also studied the problem of the Pythagorean array, and inspired by this problem, he thought of a more general problem. In 1637, he proposed a famous conjecture in the history of mathematics - Fermat's Last Theorem.

That is, when n>2, no positive integer group can be found to make the equation xn+yn=zn true. After the announcement of Fermat's Last Theorem, it attracted the attention of outstanding mathematicians from all over the world. They tenaciously explored this theorem and tried to prove it. In 1995, British mathematician Wiles finally proved Fermat's Last Theorem, solving a mystery that had puzzled countless wise men in the world for more than 300 years.

think:

As shown in the figure, in the quadrilateral ABCD, AB=3, BC=4, ∠ABC=90°, AD=12, DC=13. Use your brain! Can you find the area of ​​this quadrilateral? How to ask for it?

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Update Time: 2024-10-16

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