EntrancePythagorean theorem problem. Presentation on the topic "tasks on the Pythagorean theorem" Training tasks on the Pythagorean theorem
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“Geometry has two treasures: one of them is the Pythagorean theorem.” Johannes Kepler
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Complete the sentence:
A right triangle is a triangle whose one angle is ____ 90°
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The sides of a triangle that form a right angle are called _________ legs
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The side of a triangle opposite the right angle is called ____________ Complete the sentence: hypotenuse
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In a right triangle, the square of the hypotenuse is equal to ____________ Complete the sentence: the sum of the squares of the legs
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The proposition formulated above is called ____________ Pythagorean Theorem c² = a² + b²
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If in a triangle the square of one side is equal to the sum of the squares of the other two sides, then such a triangle is ____________ Complete the sentence: rectangular
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S=½d1 d2 S=a² S=ab S=½ah S=ah Draw lines so that the correspondence between the figure and the formula for calculating its area is correct S=½ (a +b)h S=½ ab
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Valley of Oral Problems Dunno Island Glade of Health City of Masters Fortress of Formulas Historical Path
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Valley of oral problems
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N S P 12 cm 9 cm 15 cm? Find: SP
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TO? 12 cm 13 cm N M Find: KN 5 cm
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IN? 8 cm 17 cm A D C Find: AD 15 cm
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Dunno Island
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Problem of the 12th century Indian mathematician Bhaskara “On the bank of the river a lonely poplar grew. Suddenly a gust of wind broke its trunk. The poor poplar fell. And its trunk made a right angle with the flow of the river. Remember now that in this place the river was only four feet wide, the top bent at the edge river. There are only three feet left from the trunk, I ask you, tell me soon: How tall is the poplar?”
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A car and an airplane set off from one point on earth. The car covered a distance of 8 km when the plane was at an altitude of 6 km. How far has the plane traveled in the air since takeoff? Task
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8 km 6 km? km
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Using the textbook, we solve problem No. 494 (p. 133)
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Glade of Health
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(580 - 500 BC) Pythagoras
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In order to learn science, Pythagoras traveled a lot; in one of the Greek colonies of Southern Italy in the city of Crotone, he organized a circle of young people from the aristocracy, where they were accepted with great ceremonies after long trials. Each entrant renounced his property and swore an oath to keep the founder's teachings secret. This is how the famous “Pythagorean school” arose.
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The Pythagoreans studied mathematics, philosophy, natural sciences. They made many important discoveries in arithmetic and geometry. However, there was a Decree at the school, according to which the authorship of all mathematical works was attributed to Pythagoras.
(option 1)
In rectangle ABCD, adjacent sides have a ratio of 12:5, and its diagonal is 26 cm. What is the shortest side of the rectangle?
In the parallelogram ABCD BD = 2√41 cm, AC = 26 cm, AD = 16 cm. A straight line is drawn through the intersection point of the diagonals of the parallelogram O, perpendicular to side BC. Find the segments into which this line divided side AD.
Problems on the topic “Pythagorean Theorem”
One of the outer corners right triangle is equal to 135º, and its hypotenuse is 4√2 cm. What are the sides of this triangle equal to?
The diagonals of a rhombus are 24 cm and 18 cm. What is the length of the side of the rhombus?
The major diagonal of a rectangular trapezoid is 25 cm, and the larger base is 24 cm. Find the area of the trapezoid if its smaller base is 8 cm.
The bases of an isosceles trapezoid are 10 cm and 26 cm, and the side is 17 cm. Find the area of the trapezoid.
Problems on the topic “Pythagorean Theorem”
In rectangle ABCD, adjacent sides have a ratio of 12:5, and its diagonal is 26 cm. What is the shortest side of the rectangle?
One of the external angles of a right triangle is 135º, and its hypotenuse is 4√2 cm. What are the sides of this triangle?
The diagonals of a rhombus are 24 cm and 18 cm. What is the length of the side of the rhombus?
The major diagonal of a rectangular trapezoid is 25 cm, and the larger base is 24 cm. Find the area of the trapezoid if its smaller base is 8 cm.
The bases of an isosceles trapezoid are 10 cm and 26 cm, and the side is 17 cm. Find the area of the trapezoid.
In parallelogram ABCD BD = 2√41 cm, AC = 26 cm, AD = 16 cm. A straight line is drawn through the intersection point of the diagonals of parallelogram O, perpendicular to side BC. Find the segments into which this line divided side AD.
Problems on the topic “Pythagorean Theorem”
(option 2)
6*. Two circles of radii 13 cm and 15 cm intersect. The distance between their centers O 1 and O 2 is 14 cm. The common chord of these circles AB intersects the segment O 1 O 2 at point K. Find O 1 K and KO 2 (O 1 is the center of a circle of radius 13 cm).
Problems on the topic “Pythagorean Theorem”
In rectangle ABCD, adjacent sides are in the ratio 3:4, and its diagonal is 20 cm. What is the longest side of the rectangle?
One of the external angles of a right triangle is 135º, and its hypotenuse is 5√2 cm. What are the sides of this triangle?
The diagonals of a rhombus are 12 cm and 16 cm. What is the length of the side of the rhombus?
The larger diagonal of a rectangular trapezoid is 17 cm, and the larger base is 15 cm. Find the area of the trapezoid if its smaller base is 9 cm.
5. The bases of an isosceles trapezoid are 10 cm and 24 cm, and the side is 25 cm. Find the area of the trapezoid.
Problems on the topic “Pythagorean Theorem”
In rectangle ABCD, adjacent sides are in the ratio 3:4, and its diagonal is 20 cm. What is the longest side of the rectangle?
One of the external angles of a right triangle is 135º, and its hypotenuse is 5√2 cm. What are the sides of this triangle?
The diagonals of a rhombus are 12 cm and 16 cm. What is the length of the side of the rhombus?
The larger diagonal of a rectangular trapezoid is 17 cm, and the larger base is 15 cm. Find the area of the trapezoid if its smaller base is 9 cm.
5. The bases of an isosceles trapezoid are 10 cm and 24 cm, and the side is 25 cm. Find the area of the trapezoid.
6. Two circles of radii 13 cm and 15 cm intersect. The distance between their centers O 1 and O 2 is 14 cm. The common chord of these circles AB intersects the segment O 1 O 2 at point K. Find O 1 K and KO 2 (O 1 is the center of a circle of radius 13 cm).
Entertaining problems on the topic “Pythagorean Theorem” (8th grade)
Zemlyanukhina D.V., mathematics teacher, MBOU "Anninskaya Secondary School with UIOP"
The Pythagorean theorem is rightfully considered the most important in the course of geometry and deserves close attention. It is the basis for solving many problems. Therefore, in order to develop an understanding of the significance of the Pythagorean theorem when studying both geometry and other disciplines, and the ability to apply the Pythagorean theorem to problem solving, I offer eighth-graders individual multi-level problems that require a creative approach to solution and design. Solving such entertaining problems also helps to cultivate students’ interest in the subject: mathematics no longer seems to them a dry and boring science, children see that invention, a flight of fancy, is needed here too. Creative skills.
Task No. 1. Ancient Indian problem.
Above the quiet lake
About half a foot in size
The lotus color rose.
He grew up lonely
And the wind gusts
He took it aside. No
More than a flower above the water.
The fisherman found him
Early spring
Two feet from where I grew up.
So, I'll pose a question:
“How deep is the water here?”
What is the depth in modern units of length (1 ft ≈ 0.3 m)?
Solution.
Let's make a drawing for the problem and denote the depth of the lake AC = X, then AD = AB = X + 0.5.
From the triangle ACB, according to the Pythagorean theorem, we have AB 2 – AC 2 = BC 2,
(X + 0.5) 2 – X 2 = 2 2,
X 2 + X + 0.25 – X 2 = 4,
The depth of the lake is therefore 3.75 feet.
3.75 ∙ 0.3 = 1.125 (m)
Answer: 3.75 feet or 1.125 m.
Task No. 2. The problem of an Indian mathematician of the 12th century. Bhaskars.
A lonely poplar grew on the bank of the river. Suddenly a gust of wind broke its trunk. The poor poplar fell. And its trunk made a right angle with the flow of the river. Remember now that in that place the river was only four feet wide. The top leaned over the edge of the river, only three feet of the trunk remained. I ask you, tell me soon: how tall is the poplar?
Solution.
Answer: 8 feet.
Task No. 3. Arab mathematician's problem XI V.
There is a palm tree growing on both banks of the river, one opposite the other. The height of one is 30 cubits, the other is 20 cubits. The distance between their bases is 50 cubits. A bird sits on the top of each palm tree. Suddenly both birds noticed a fish swimming to the surface of the water between the palm trees. They rushed towards her at once and reached her at the same time. At what distance from the base of the taller palm did the fish appear?
Task No. 4. Egyptian problem.
At a depth of 12 feet there is a lotus with a 13-foot stem. Determine how far the flower can deviate from the vertical passing through the point of attachment of the stem to the bottom.
Solution.
Answer: 5 feet.
Task No. 5.
A bamboo trunk 9 feet high was broken by a storm so that if the top of it was bent to the ground, the top would touch the ground at a distance of 3 feet from the base of the trunk. At what height was the trunk broken?
Solution.
Answer: 4 feet.
Task No. 6.
In the center of a square pond, 10 feet long and wide, reeds grow one foot above the surface of the water. If you bend it to the shore, to the middle of the side of the pond, then its top will reach the shore. What is the depth of the pond in modern units of length (1 ft ≈ 0.3 m)?
Solution.
Let us denote the depth of lake B D = x, then AB = BC = x + 1 – the length of the reed. From ∆ВDC according to the Pythagorean theorem CD 2 = CB 2 –ВD 2,
5 2 = (x + 1) 2 – x 2,
25 = x 2 + 2x + 1 – x 2,
So the depth of the pond is 12 feet. 12 ∙ 0.3 = 3.6 (m).
Answer: 3.6 m.
Task No. 7.
The subway escalator has 17 steps from the ground concourse floor to the underground station floor. The width of the steps is 40 cm, the height is 20 cm. Determine a) the length of the stairs, b) the vertical depth of the station.
Solution.
|
| a) Let AB be the length of a staircase of 17 steps. From ∆AK D by Pythagorean theorem AD= (cm), AB = 45 ∙ 17 = 765 (cm) = 7.65 (m). b) BC = 40 ∙ 17 = 680 (cm). From ∆ASV according to the Pythagorean theorem AC= (cm) = = 3.5 (m). |
Answer: the length of the stairs is 7.65 m, the depth of the station is 3.5 m.
Task No. 8.
Parallel to the straight road at a distance of 500m from it there is a chain of riflemen. The distance between the extreme arrows is 120 m, the flight range of the bullet is 2800 m. Which section of the road is under fire?
Solution.
|
| From ∆AN D by Pythagorean theorem AN= (km), AB = 2 ∙ AN + NK, AB = 2 ∙ 2.755 + 0.12 ≈ 5.63 (km). |
Answer: 5.63 km.
Task No. 9.
The swimmer swam from the river bank, rowing all the time in a direction perpendicular to the shore (we consider the river banks to be parallel). He swam, approaching the opposite shore at a speed of 3 km/h. After 5 min. he was on the opposite bank. Find out at what distance from the start of the swim he came out on the opposite bank, considering the speed of the current everywhere equal to 6 km/h.
Solution.
|
| The swimmer was approaching the opposite shore at a speed AB = 50 ∙ 5 = 250 (m). River flow speed AC = ≈ 250 ∙ 2.24=560 (m) |
, means the width of the river
, therefore, the current carried it away in 5 minutes. at 500m (BC=500m). Using the Pythagorean theorem, we find the distance from the point of the initial swim to the point of exit to the opposite shoreAnswer: 560 m.
Task No. 10.
You are sailing a boat on a lake and want to know its depth. Is it possible to use the reeds sticking out of the water for this without tearing them out?
Solution.
|
| Slightly tilting the reed and holding it taut, measure the distanceA between points A and B, at which the reeds cross the surface of the water in a vertical and inclined position, respectively. Return the reed to its original position and determine the height V above the water, to which point B of the inclined reed will rise, taking the initial position C. Then denoting the base of the reed by D, and by X – the desired depth AD, from the rectangular ∆АВD we find using the Pythagorean theorem X 2 +a 2 = (x+b) 2 , X 2 +a 2 = x 2 +2хв+в 2 2хв=а 2 -V 2 , x= |
Task No. 11.
How far can you see from a lighthouse of a given height above sea level?
Solution.
Answer: from a height of 125 m from the lighthouse, a distance of 40 km is visible.
Task No. 12.
The helicopter rises vertically at a speed of 4 m/s. Determine the speed of the helicopter if the speed of the wind blowing horizontally is 3 m/s.
Solution.
v 2 = 3 2 + 4 2 = 25
Answer: 5 m/s.
Literature:
Borisova N.A. Lesson-conference on geometry in 8th grade
Municipal budgetary educational institution
"Krasnikovskaya basic secondary school"
Znamensky district, Oryol region
Lesson summary on the topic:
“Solving problems on the topic: “The Pythagorean Chamber”
Mathematic teacher -
Filina Marina Alexandrovna
2015 – 2016 academic year
Solving problems on the topic: “The Pythagorean Chamber”
The purpose of the lesson:
- Strengthen the ability to apply the Pythagorean theorem when solving problems
- Develop logical thinking
- Learn to use the acquired knowledge in practice and in everyday life
Lesson type: lesson of generalization and consolidation of the studied material.
Forms of work in the lesson:frontal, individual, independent.
Equipment: computer; multimedia projector; presentation for the lesson.
During the classes
1. Organizational moment
Greeting, checking readiness for the lesson (workbooks, textbooks, writing materials).
Mathematical dictation
- Which triangle is called a right triangle?
- What is the sum of the angles of a right triangle?
- What is the sum of the acute angles in a right triangle?
- Formulate the property of a leg lying opposite an angle of 30 degrees.
- State the Pythagorean theorem.
- What is the opposite side called? right angle?
- What is the side adjacent to a right angle called?
Checking the mathematical dictation
- If there is a right angle.
- 180°
- 3. 90°
4. Leg of a right triangle lying opposite the angle
At 30° it is equal to half the hypotenuse.
5. In a right triangle, the square of the hypotenuse
Equal to the sum of the squares of the legs.
6. Hypotenuse.
7. Leg.
Problem solving
No. 2. How far should the lower end of the ladder be moved from the wall of the house?
Which length is 13 m so that its upper end is at a height of 12 m?
No. 3. Given:
∆ABC isosceles
AB = 13 cm,
ID – height, ID=12 cm
Find: AC
№ 4.
Given: ABCD – rhombus,
AC, VD – diagonals,
AC = 12 cm, BD = 16 cm.
Find: P ABCD
Physical education pause
Test
1. Which scientist’s theorem did we use today in class?
a) Democritus; b) Magnitsky; c) Pythagoras; d) Lomonosov.
2. What did this mathematician discover?
a) theorem; b) manuscript; c) an ancient temple; d) task.
3. What is the largest side called in a right triangle?
a) median; b) leg; c) bisector; d) hypotenuse.
4. Why was the theorem called the “bride’s theorem”
a) because it was written for the bride;
b) because it was written by the bride;
c) because the drawing looks like a “butterfly”, and “butterfly” is translated as “nymph” or “bride”;
d) because it is a mysterious theorem.
5. Why the theorem was called the “bridge of donkeys”
a) it was used for training donkeys;
b) only the smart and stubborn could overcome this bridge and prove this theorem;
c) it was written by “donkeys”;
d) a very complex proof of the theorem.
6. In the Pythagorean theorem, the square of the hypotenuse is equal to
a) the sum of the lengths of the sides of a triangle;
b) the sum of the squares of the legs;
c) area of the triangle;
d) area of the square.
7. What are the sides of the Egyptian triangle?
a) 1, 2, 3; b) 3,4,5; c)2,3,4; d) 6,7,8.
Lesson summary, grading.
Homework - № 9, № 12
Reflexions
“I repeated...” “I found out...”
“I have consolidated...” “I have learned to decide...”
"I like it…"
Lesson topic
Pythagorean theorem
Lesson Objectives
To introduce schoolchildren to the Pythagorean theorem;
Formulate and prove the Pythagorean theorem;
Introduce schoolchildren to different methods of applying this theorem when solving problems;
Develop skills to use acquired knowledge in practice;
To develop students’ attention, independence and interest in geometry;
Foster a culture of mathematical speech.
Lesson Objectives
Learn to use the properties of shapes when completing tasks.
Be able to apply the Pythagorean theorem when solving problems.
Lesson Plan
Brief biographical information.
Theorem and its proof.
Interesting Facts.
Problem solving.
Homework.
Brief biographical information about Pythagoras
Unfortunately, Pythagoras did not leave any writings about his biography, so we can only learn all the information about this great philosopher and famous mathematician through the memories of his followers, and even then they are not always fair. Therefore, there are many legends about this man. But the truth is that Pythagoras was a great Hellenic sage, philosopher and talented mathematician.
According to unreliable information, the great sage and brilliant scientist Pythagoras was born into a far from poor family, on the island of Samosea, around 570 BC.
The birth of a brilliant child was predicted by Paphia. Therefore, the future luminary received his name Pythagoras, which means that this is exactly the one whom Paphia announced. She predicted that the born baby would bring a lot of benefit and goodness to people in the future.
The newborn was incredibly beautiful, and in time he pleased those around him with his outstanding abilities. And since the young talent whiled away his days among the wise elders, this bore fruit in the future. This is how, thanks to Hermodamantus, Pythagoras fell in love with music, and Pherecydes directed the child’s mind to logos. After living in Samosea, Pythagoras went to Miletus, where he met another scientist - Thales.
Pythagoras became acquainted with the knowledge of all the sages known at that time, since he was allowed to study and learn all the sacraments that were forbidden to others. He tried to get to the bottom of the truth and absorb all the knowledge accumulated by mankind.
After twenty-two years in Egypt, Pythagoras moved to Babylon, where he continued his communication with various sages and magicians. Returning at the end of his life to Samios, he was recognized as one of the wisest people that time.
Pythagorean theorem
Even a person who has not yet had the opportunity to study this theorem has probably heard the statement about “Pythagorean pants.” The peculiarity of this theorem is that it has become one of the key theorems of Euclidean geometry. It makes it easy to find and establish correspondence between the sides of a right triangle.
The Pythagorean theorem was remembered by every schoolchild not only for the statement: “Pythagorean pants are equal on all sides,” but for its simplicity and significance. And at first glance, this theorem, although it seems simple, has great importance, since in geometry it is applied at virtually every step.
The Pythagorean theorem has a large number of different proofs and is probably the only theorem that has such a huge number of proofs. This diversity underscores the limitless significance of this theorem.
The Pythagorean theorem contains geometric, algebraic, mechanical and other proofs.
There are many different legends about the discovery of the theorem by Pythagoras. But, despite all this, the name of Pythagoras forever entered the history of geometry and firmly merged with the Pythagorean theorem. After all, this brilliant mathematician will be the first to present a proof of the theorem that bears his name.
Statements of the theorem
There are several formulations of the Pythagorean theorem.
The Euclidean Theorem tells us that the square of the side of a right triangle drawn over its right angle is equal to the squares on the sides enclosing the right angle.
Assignment: Find different formulations of the Pythagorean theorem. Do you find any difference in them?
Simplified Euclid's proof
Regardless of whether we take the decomposition method or the Euclidean proof, any arrangement of squares can be used. In some cases, minor simplifications can be achieved.
Let's take a square, which is built on one of the legs and has the same location as the triangle. We see that the continuation of the side opposite to the leg of this square passes through the vertex of the square, which is built on the hypotenuse.
The proof of the theorem looks quite simple, since it will be enough to simply compare the areas of the figures with the area of the triangle. And we see that S of a triangle is equal to ½ the area of a square, and also ½ S of a rectangle.
The simplest proof
Algebraic proof
The algebraic proof of the Pythagorean theorem includes elementary methods that are present in algebra. These are methods of solving equations combined with a method of changing variables.
Let's look at this evidence in more detail. And so, we have a rectangle ABC, whose right angle is C.
Draw the CD height from this corner.
According to the definition of cosine of an angle, we get:
cosA=AD/AC=AC/AB. Hence AB*AD=AC2.
And correspondingly:
cosB = BD/BC=BC/AB.
Hence AB*BD=BC2.
Now let’s add these equalities term by term and see that: AD+DB=AB,
AC2+BC2=AB(AD+DB)=AB2.
That's all, the theorem is proven.
Scientists “proved” the Pythagorean theorem with the help of cartoons. A group of like-minded people from the Institute. Steklova received a prize for an original mathematical project that they developed for schoolchildren and teachers. They created mini lessons in mathematics that turned this boring subject into a very interesting and educational one. The young scientists released their unusual sketches on discs and posted them on the Internet for public viewing.
Questions
1. Who is Pythagoras?
2. What does the Pythagorean theorem say?
3. What are the formulations of the Pythagorean theorem?
4. When solving what problems is the Pythagorean theorem used?
5. Where did the Pythagorean theorem find practical application?
6. What ways do you know of using the Pythagorean theorem?
Problems using the Pythagorean theorem
Using your knowledge of the Pythagorean theorem, try to solve the following problems:
Two groups of tourists left the tourist base at the same time. The first group went south and walked seven kilometers, and the second turned west and walked nine kilometers. Using knowledge of the theorem, find the distance between groups of tourists.
If in a right triangle its leg is 15 cm and the hypotenuse is 16 cm, then what will the second leg be equal to?
What will be the area of the trapezoid when its major base is 24 cm, its smaller base is 16, and the major diagonal of a rectangular trapezoid is 26 cm?
Homework
Present in the form of a short report several proofs of the Pythagorean theorem that you understand and solve the problems.
1. Find the diagonal of a right triangle, provided that its sides are 8 cm and 32 cm.
2. Find the median of the triangle, which is drawn to the base, if in isosceles triangle the perimeter is 38 cm, and its side is 15 cm.
3. A triangle has sides equal to 10 cm, 6 cm and 9 cm. Try to determine whether this triangle is right-angled?