Rectangular
triangles
Geometry, 7th grade
To the textbook by L.S. Atanasyan
mathematics teacher of the highest category
Municipal educational institution "Upshinskaya basic secondary school"
Orsha district of the Republic of Mari El
AC, BC – legs
AB - hypotenuse
Property 1 0 . The sum of the acute angles of a right triangle is 90 0.
Task 1. Find angle A of right triangle ABC with right angle C if: a) ے B = 32 0; b) ے B is 2 times less than angle A; c) ے B is 20 0 less than angle A.
Task 2.
Task 3.
Angle A:
BC – leg lying opposite angle A
AC – leg adjacent to angle A
Angle B:
AC - leg, ...
BC - leg, ...
Name the legs opposite angles N and K
Name the legs adjacent to angles N and K
0
Task. Prove that 0 , is equal to half the hypotenuse.
Property 2 0 . A leg of a right triangle lying opposite an angle of 30 0 , is equal to half the hypotenuse.
Right triangle with angle 30 0
Task. Prove that 0 .
Property 3 0 . If a leg of a right triangle is equal to half the hypotenuse, then the angle opposite this leg is 30 0 .
Right triangle with angle 30 0
Problem 4 .
AB = 12 cm. Find BC
Task 5.
BC = 7.5 cm. Find AB
Task 6.
AB + BC = 15 cm.
Find AB and BC
Right triangle with angle 30 0
Task 7.
AC = 4 cm. Find AB
Task 8.
AB - AC = 15 cm.
Find AB and AC
Right triangle with angle 30 0
Problem 9 .
Find the acute angles of right triangle ABC if AB = 12 cm, BC = 6 cm.
Right triangle with angle 30 0
Problem 10 .
Find the acute angles of a right triangle if the angle between the bisector and the altitude drawn from the vertex right angle, is equal to 15 0.
SC - bisector
CM - height
Right triangle with angle 30 0
Problem 11 .
In an isosceles triangle, one of the angles is 120 0 and the base is 4 cm. Find the height drawn to the side.
AM - height
Right triangle with angle 30 0
Problem 12 .
The altitude drawn to the lateral side of an isosceles triangle bisects the angle between the base and the bisector. Find the angles of an isosceles triangle.
AK – bisector of angle A
AM - height
Right triangle with angle 30 0
Problem 14 .
Prove that if the triangle is right-angled, then the median drawn from the vertex of the right angle is equal to half the hypotenuse.
Property 4 0 .
ΔАВС - rectangular
SM – median
We got a contradiction!
Right triangle with angle 30 0
Problem 13 .
Prove that if the median of a triangle is equal to half the side to which it is drawn, then the triangle is right-angled.
VM – median
Prove: ΔABC - rectangular
Property 5 0 .
Some properties of right triangles
Property 1 0 . The sum of the acute angles of a right triangle is 90 0 .
Property 2 0 . A leg of a right triangle lying opposite an angle of 30 0 , equal to half the hypotenuse .
Property 3 0 . If a leg of a right triangle is equal to half the hypotenuse, then the angle opposite this leg is 30 0 .
Property 4 0 . In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse.
Property 5 0 . If the median of a triangle is equal to half the side to which it is drawn, then this triangle is right-angled.
Properties of a right triangle
Dear seventh graders, you already know what geometric shapes are called triangles, you know how to prove signs of their equality. You also know about special cases of triangles: isosceles and right angles. You are well aware of the properties of isosceles triangles.
But right triangles also have many properties. One obvious one has to do with the triangle interior angle sum theorem: in a right triangle, the sum of the acute angles is 90°. You will learn the most amazing property of a right triangle in 8th grade when you study the famous Pythagorean theorem.
Now we will talk about two more important properties. One is for 30° right triangles and the other is for random right triangles. Let us formulate and prove these properties.
You are well aware that in geometry it is customary to formulate statements that are converse to proven ones, when the condition and conclusion in the statement change places. Converse statements are not always true. In our case, both converse statements are true.
Property 1.1 In a right triangle, the leg opposite the 30° angle is equal to half the hypotenuse.
Proof: Consider the rectangular ∆ ABC, in which ÐA=90°, ÐB=30°, then ÐC=60°..gif" width="167" height="41">, therefore, what needed to be proven.
Property 1.2 (reverse to property 1.1) If in a right triangle the leg is equal to half the hypotenuse, then the angle opposite it is 30°.
Property 2.1 In a right triangle, the median drawn to the hypotenuse is equal to half the hypotenuse.
Let's consider a rectangular ∆ ABC, in which РВ=90°.
BD-median, that is, AD=DC. Let's prove that .
To prove this, we will make an additional construction: we will continue BD beyond point D so that BD=DN and connect N with A and C..gif" width="616" height="372 src=">
Given: ∆ABC, ÐC=90o, ÐA=30o, ÐBEC=60o, EC=7cm
1. ÐEBC=30o, because in a rectangular ∆BCE the sum of acute angles is 90o
2. BE=14cm(property 1)
3. ÐABE=30o, since ÐA+ÐABE=ÐBEC (property of the external angle of a triangle) therefore ∆AEB is isosceles AE=EB=14cm.
BC=2AN=20 cm (property 2).
Task 3. Prove that the altitude and median of a right triangle taken to the hypotenuse form an angle equal to the difference between the acute angles of the triangle.
Given: ∆ ABC, ÐBAC=90°, AM-median, AH-height.
Prove: RMAN=RS-RV.
Proof:
1)РМАС=РС (by property 2 ∆ AMC-isosceles, AM=SM)
2) ÐMAN = ÐMAS-ÐNAS = ÐS-ÐNAS.
It remains to prove that РНАС=РВ. This follows from the fact that ÐB+ÐC=90° (in ∆ ABC) and ÐNAS+ÐC=90° (from ∆ ANS).
So, RMAN = RS-RV, which is what needed to be proven.
https://pandia.ru/text/80/358/images/image014_39.gif" width="194" height="184">Given: ∆ABC, ÐBAC=90°, AN-height, .
Find: РВ, РС.
Solution: Let's take the median AM. Let AN=x, then BC=4x and
VM=MS=AM=2x.
In a rectangular ∆AMN, the hypotenuse AM is 2 times larger than the leg AN, therefore ÐAMN=30°. Since VM=AM,
РВ=РВAM100%">
Let's extend AC beyond point A so that AD=AC. Then ∆ABC=∆ABD (on 2 legs). BD=BC=2AC=CD, thus ∆DBC-equilateral, ÐC=60o and ÐABC=30o.
Problem 5
In an isosceles triangle, one of the angles is 120°, the base is 10 cm. Find the height drawn to the side.
Solution: to begin with, we note that the angle of 120° can only be at the vertex of the triangle and that the height drawn to the side will fall on its continuation.
AB - staircase, K - kitten.
In any position of the ladder, until it finally falls to the ground, ∆ABC is rectangular. MC - median ∆ABC.
According to property 2 SC = 1/2AB. That is, at any moment in time the length of the segment SK is constant.
Answer: point K will move along a circular arc with center C and radius CK=1/2AB.
Problems for independent solution.
One of the angles of a right triangle is 60°, and the difference between the hypotenuse and the shorter leg is 4 cm. find the length of the hypotenuse. In a rectangular ∆ ABC with hypotenuse BC and angle B equal to 60°, the height AD is drawn. Find DC if DB=2cm. B ∆ABC ÐC=90o, CD - height, BC=2ВD. Prove that AD=3ВD. The height of a right triangle divides the hypotenuse into parts 3 cm and 9 cm. Find the angles of the triangle and the distance from the middle of the hypotenuse to the longer leg. The bisector splits the triangle into two isosceles triangles. Find the angles of the original triangle. The median splits the triangle into two isosceles triangles. Is it possible to find angles
The original triangle?
Intermediate level
Right triangle. The Complete Illustrated Guide (2019)
RECTANGULAR TRIANGLE. ENTRY LEVEL.
In problems, the right angle is not at all necessary - the lower left, so you need to learn to recognize a right triangle in this form,
and in this
and in this 
What's good about a right triangle? Well... first of all, there are special beautiful names for his sides.
Attention to the drawing!
Remember and don't confuse: there are two legs, and there is only one hypotenuse(one and only, unique and longest)!
Well, we’ve discussed the names, now the most important thing: the Pythagorean Theorem.
Pythagorean theorem.
This theorem is the key to solving many problems involving a right triangle. It was proved by Pythagoras in completely immemorial times, and since then it has brought a lot of benefit to those who know it. And the best thing about it is that it is simple.
So, Pythagorean theorem:
Do you remember the joke: “Pythagorean pants are equal on all sides!”?
Let's draw these same Pythagorean pants and look at them. 
Doesn't it look like some kind of shorts? Well, on which sides and where are they equal? Why and where did the joke come from? And this joke is connected precisely with the Pythagorean theorem, or more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:
"Sum areas of squares, built on the legs, is equal to square area, built on the hypotenuse."
Does it really sound a little different? And so, when Pythagoras drew the statement of his theorem, this is exactly the picture that came out.
In this picture, the sum of the areas of the small squares is equal to the area of the large square. And so that children can better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty came up with this joke about Pythagorean pants.
Why are we now formulating the Pythagorean theorem?
Did Pythagoras suffer and talk about squares?
You see, in ancient times there was no... algebra! There were no signs and so on. There were no inscriptions. Can you imagine how terrible it was for the poor ancient students to remember everything in words??! And we can rejoice that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to remember it better:
It should be easy now:
| Square of the hypotenuse equal to the sum squares of legs. |
Well, the most important theorem about right triangles has been discussed. If you are interested in how it is proven, read the following levels of theory, and now let's move on... to dark forest... trigonometry! To the terrible words sine, cosine, tangent and cotangent.
Sine, cosine, tangent, cotangent in a right triangle.
In fact, everything is not so scary at all. Of course, the “real” definition of sine, cosine, tangent and cotangent should be looked at in the article. But I really don’t want to, do I? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:
Why is everything just about the corner? Where is the corner? In order to understand this, you need to know how statements 1 - 4 are written in words. Look, understand and remember!
1.
Actually it sounds like this:
What about the angle? Is there a leg that is opposite the corner, that is, an opposite (for an angle) leg? Of course there is! This is a leg!
What about the angle? Look carefully. Which leg is adjacent to the corner? Of course, the leg. This means that for the angle the leg is adjacent, and
Now, pay attention! Look what we got:
See how cool it is:
Now let's move on to tangent and cotangent.
How can I write this down in words now? What is the leg in relation to the angle? Opposite, of course - it “lies” opposite the corner. What about the leg? Adjacent to the corner. So what have we got?
See how the numerator and denominator have swapped places?
And now the corners again and made an exchange:
Resume
Let's briefly write down everything we've learned.
 |
Pythagorean theorem: |
The main theorem about right triangles is the Pythagorean theorem.
Pythagorean theorem
By the way, do you remember well what legs and hypotenuse are? If not very good, then look at the picture - refresh your knowledge
It is quite possible that you have already used the Pythagorean theorem many times, but have you ever wondered why such a theorem is true? How can I prove it? Let's do like the ancient Greeks. Let's draw a square with a side.
See how cleverly we divided its sides into lengths and!
Now let's connect the marked dots
Here we, however, noted something else, but you yourself look at the drawing and think why this is so.
What is the area of the larger square? Right, . What about a smaller area? Certainly, . The total area of the four corners remains. Imagine that we took them two at a time and leaned them against each other with their hypotenuses. What happened? Two rectangles. This means that the area of the “cuts” is equal.
Let's put it all together now.
Let's transform:
So we visited Pythagoras - we proved his theorem in an ancient way.
Right triangle and trigonometry
For a right triangle, the following relations hold:
Sinus acute angle equal to the ratio opposite side to the hypotenuse
The cosine of an acute angle is equal to the ratio of the adjacent leg to the hypotenuse.
The tangent of an acute angle is equal to the ratio of the opposite side to the adjacent side.
The cotangent of an acute angle is equal to the ratio of the adjacent side to the opposite side.
And once again all this in the form of a tablet:
It's very convenient!
Signs of equality of right triangles
I. On two sides
II. By leg and hypotenuse
III. By hypotenuse and acute angle
IV. Along the leg and acute angle
a) 
b) 
Attention! It is very important here that the legs are “appropriate”. For example, if it goes like this:
THEN TRIANGLES ARE NOT EQUAL, despite the fact that they have one identical acute angle.
It is necessary that in both triangles the leg was adjacent, or in both it was opposite.
Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles? Look at the topic “and pay attention to the fact that for equality of “ordinary” triangles, three of their elements must be equal: two sides and the angle between them, two angles and the side between them, or three sides. But for the equality of right triangles, only two corresponding elements are enough. Great, right?
The situation is approximately the same with the signs of similarity of right triangles.
Signs of similarity of right triangles
I. Along an acute angle
II. On two sides
III. By leg and hypotenuse
Median in a right triangle
Why is this so?
Instead of a right triangle, consider a whole rectangle.
Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What is known about the diagonals of a rectangle?
And what follows from this?
So it turned out that
- - median:
Remember this fact! Helps a lot!
What’s even more surprising is that the opposite is also true.
What good can be obtained from the fact that the median drawn to the hypotenuse is equal to half the hypotenuse? Let's look at the picture
Look carefully. We have: , that is, the distances from the point to all three vertices of the triangle turned out to be equal. But there is only one point in the triangle, the distances from which from all three vertices of the triangle are equal, and this is the CENTER OF THE CIRCLE. So what happened?
So let's start with this “besides...”.
Let's look at and.
But similar triangles have all equal angles!
The same can be said about and
Now let's draw it together:
What benefit can be derived from this “triple” similarity?
Well, for example - two formulas for the height of a right triangle.
Let us write down the relations of the corresponding parties:
To find the height, we solve the proportion and get the first formula "Height in a right triangle":
So, let's apply the similarity: .
What will happen now?
Again we solve the proportion and get the second formula:
You need to remember both of these formulas very well and use the one that is more convenient. Let's write them down again
Pythagorean theorem:
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: .
Signs of equality of right triangles:
- on two sides:
- by leg and hypotenuse: or
- along the leg and adjacent acute angle: or
- along the leg and the opposite acute angle: or
- by hypotenuse and acute angle: or.
Signs of similarity of right triangles:
- one acute corner: or
- from the proportionality of two legs:
- from the proportionality of the leg and hypotenuse: or.
Sine, cosine, tangent, cotangent in a right triangle
- The sine of an acute angle of a right triangle is the ratio of the opposite side to the hypotenuse:
- The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
- The tangent of an acute angle of a right triangle is the ratio of the opposite side to the adjacent side:
- The cotangent of an acute angle of a right triangle is the ratio of the adjacent side to the opposite side: .
Height of a right triangle: or.
In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse: .
Area of a right triangle:
- via legs:
Intermediate level
Right triangle. The Complete Illustrated Guide (2019)
RECTANGULAR TRIANGLE. ENTRY LEVEL.
In problems, the right angle is not at all necessary - the lower left, so you need to learn to recognize a right triangle in this form,
and in this
and in this
What's good about a right triangle? Well..., firstly, there are special beautiful names for its sides.
Attention to the drawing!
Remember and don't confuse: there are two legs, and there is only one hypotenuse(one and only, unique and longest)!
Well, we’ve discussed the names, now the most important thing: the Pythagorean Theorem.
Pythagorean theorem.
This theorem is the key to solving many problems involving a right triangle. It was proved by Pythagoras in completely immemorial times, and since then it has brought a lot of benefit to those who know it. And the best thing about it is that it is simple.
So, Pythagorean theorem:
Do you remember the joke: “Pythagorean pants are equal on all sides!”?
Let's draw these same Pythagorean pants and look at them.
Doesn't it look like some kind of shorts? Well, on which sides and where are they equal? Why and where did the joke come from? And this joke is connected precisely with the Pythagorean theorem, or more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:
"Sum areas of squares, built on the legs, is equal to square area, built on the hypotenuse."
Does it really sound a little different? And so, when Pythagoras drew the statement of his theorem, this is exactly the picture that came out.
In this picture, the sum of the areas of the small squares is equal to the area of the large square. And so that children can better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty came up with this joke about Pythagorean pants.
Why are we now formulating the Pythagorean theorem?
Did Pythagoras suffer and talk about squares?
You see, in ancient times there was no... algebra! There were no signs and so on. There were no inscriptions. Can you imagine how terrible it was for the poor ancient students to remember everything in words??! And we can rejoice that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to remember it better:
It should be easy now:
| The square of the hypotenuse is equal to the sum of the squares of the legs. |
Well, the most important theorem about right triangles has been discussed. If you are interested in how it is proven, read the following levels of theory, and now let's go further... into the dark forest... trigonometry! To the terrible words sine, cosine, tangent and cotangent.
Sine, cosine, tangent, cotangent in a right triangle.
In fact, everything is not so scary at all. Of course, the “real” definition of sine, cosine, tangent and cotangent should be looked at in the article. But I really don’t want to, do I? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:
Why is everything just about the corner? Where is the corner? In order to understand this, you need to know how statements 1 - 4 are written in words. Look, understand and remember!
1.
Actually it sounds like this:
What about the angle? Is there a leg that is opposite the corner, that is, an opposite (for an angle) leg? Of course there is! This is a leg!
What about the angle? Look carefully. Which leg is adjacent to the corner? Of course, the leg. This means that for the angle the leg is adjacent, and
Now, pay attention! Look what we got:
See how cool it is:
Now let's move on to tangent and cotangent.
How can I write this down in words now? What is the leg in relation to the angle? Opposite, of course - it “lies” opposite the corner. What about the leg? Adjacent to the corner. So what have we got?
See how the numerator and denominator have swapped places?
And now the corners again and made an exchange:
Resume
Let's briefly write down everything we've learned.
|
Pythagorean theorem: |
The main theorem about right triangles is the Pythagorean theorem.
Pythagorean theorem
By the way, do you remember well what legs and hypotenuse are? If not very good, then look at the picture - refresh your knowledge
It is quite possible that you have already used the Pythagorean theorem many times, but have you ever wondered why such a theorem is true? How can I prove it? Let's do like the ancient Greeks. Let's draw a square with a side.
See how cleverly we divided its sides into lengths and!
Now let's connect the marked dots
Here we, however, noted something else, but you yourself look at the drawing and think why this is so.
What is the area of the larger square? Right, . What about a smaller area? Certainly, . The total area of the four corners remains. Imagine that we took them two at a time and leaned them against each other with their hypotenuses. What happened? Two rectangles. This means that the area of the “cuts” is equal.
Let's put it all together now.
Let's transform:
So we visited Pythagoras - we proved his theorem in an ancient way.
Right triangle and trigonometry
For a right triangle, the following relations hold:
The sine of an acute angle is equal to the ratio of the opposite side to the hypotenuse
The cosine of an acute angle is equal to the ratio of the adjacent leg to the hypotenuse.
The tangent of an acute angle is equal to the ratio of the opposite side to the adjacent side.
The cotangent of an acute angle is equal to the ratio of the adjacent side to the opposite side.
And once again all this in the form of a tablet:
It's very convenient!
Signs of equality of right triangles
I. On two sides
II. By leg and hypotenuse
III. By hypotenuse and acute angle
IV. Along the leg and acute angle
a)
b)
Attention! It is very important here that the legs are “appropriate”. For example, if it goes like this:
THEN TRIANGLES ARE NOT EQUAL, despite the fact that they have one identical acute angle.
It is necessary that in both triangles the leg was adjacent, or in both it was opposite.
Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles? Look at the topic “and pay attention to the fact that for equality of “ordinary” triangles, three of their elements must be equal: two sides and the angle between them, two angles and the side between them, or three sides. But for the equality of right triangles, only two corresponding elements are enough. Great, right?
The situation is approximately the same with the signs of similarity of right triangles.
Signs of similarity of right triangles
I. Along an acute angle
II. On two sides
III. By leg and hypotenuse
Median in a right triangle
Why is this so?
Instead of a right triangle, consider a whole rectangle.
Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What is known about the diagonals of a rectangle?
And what follows from this?
So it turned out that
- - median:
Remember this fact! Helps a lot!
What’s even more surprising is that the opposite is also true.
What good can be obtained from the fact that the median drawn to the hypotenuse is equal to half the hypotenuse? Let's look at the picture
Look carefully. We have: , that is, the distances from the point to all three vertices of the triangle turned out to be equal. But there is only one point in the triangle, the distances from which from all three vertices of the triangle are equal, and this is the CENTER OF THE CIRCLE. So what happened?
So let's start with this “besides...”.
Let's look at and.
But similar triangles have all equal angles!
The same can be said about and
Now let's draw it together:
What benefit can be derived from this “triple” similarity?
Well, for example - two formulas for the height of a right triangle.
Let us write down the relations of the corresponding parties:
To find the height, we solve the proportion and get the first formula "Height in a right triangle":
So, let's apply the similarity: .
What will happen now?
Again we solve the proportion and get the second formula:
You need to remember both of these formulas very well and use the one that is more convenient. Let's write them down again
Pythagorean theorem:
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: .
Signs of equality of right triangles:
- on two sides:
- by leg and hypotenuse: or
- along the leg and adjacent acute angle: or
- along the leg and the opposite acute angle: or
- by hypotenuse and acute angle: or.
Signs of similarity of right triangles:
- one acute corner: or
- from the proportionality of two legs:
- from the proportionality of the leg and hypotenuse: or.
Sine, cosine, tangent, cotangent in a right triangle
- The sine of an acute angle of a right triangle is the ratio of the opposite side to the hypotenuse:
- The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
- The tangent of an acute angle of a right triangle is the ratio of the opposite side to the adjacent side:
- The cotangent of an acute angle of a right triangle is the ratio of the adjacent side to the opposite side: .
Height of a right triangle: or.
In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse: .
Area of a right triangle:
- via legs: