This lesson will cover the topic “Isosceles triangle and its properties.” You will learn what isosceles and equilateral triangles look like and how they are characterized. Prove the theorem on the equality of angles at the base of an isosceles triangle. Consider also the theorem about the bisector (median and altitude) drawn to the base of an isosceles triangle. At the end of the lesson, you will solve two problems using the definition and properties of an isosceles triangle.
Definition:Isosceles is called a triangle whose two sides are equal.
Rice. 1. Isosceles triangle
AB = AC - sides. BC - foundation.
The area of an isosceles triangle is equal to half the product of its base and height.
Definition:Equilateral is called a triangle in which all three sides are equal.
Rice. 2. Equilateral triangle
AB = BC = SA.
Theorem 1: In an isosceles triangle, the base angles are equal.
Given: AB = AC.
Prove:∠B =∠C.
Rice. 3. Drawing for the theorem
Proof: triangle ABC = triangle ACB according to the first sign (two equal sides and the angle between them). From the equality of triangles it follows that all corresponding elements are equal. This means ∠B = ∠C, which is what needed to be proven.
Theorem 2: In an isosceles triangle bisector drawn to the base is median And height.
Given: AB = AC, ∠1 = ∠2.
Prove:ВD = DC, AD perpendicular to BC.
Rice. 4. Drawing for Theorem 2
Proof: triangle ADB = triangle ADC according to the first sign (AD - general, AB = AC by condition, ∠BAD = ∠DAC). From the equality of triangles it follows that all corresponding elements are equal. BD = DC since they lie opposite equal angles. So AD is the median. Also ∠3 = ∠4, since they lie opposite equal sides. But, besides, they are equal in total. Therefore, ∠3 = ∠4 = . This means that AD is the height of the triangle, which is what we needed to prove.
In the only case a = b = . In this case, the lines AC and BD are called perpendicular.
Since the bisector, height and median are the same segment, the following statements are also true:
The altitude of an isosceles triangle drawn to the base is the median and bisector.
The median of an isosceles triangle drawn to the base is the altitude and bisector.
Example 1: In an isosceles triangle, the base is half the size of the side, and the perimeter is 50 cm. Find the sides of the triangle.
Given: AB = AC, BC = AC. P = 50 cm.
Find: BC, AC, AB.
Solution:
Rice. 5. Drawing for example 1
Let us denote the base BC as a, then AB = AC = 2a.
2a + 2a + a = 50.
5a = 50, a = 10.
Answer: BC = 10 cm, AC = AB = 20 cm.
Example 2: Prove that in an equilateral triangle all angles are equal.
Given: AB = BC = SA.
Prove:∠A = ∠B = ∠C.
Proof:
Rice. 6. Drawing for example
∠B = ∠C, since AB = AC, and ∠A = ∠B, since AC = BC.
Therefore, ∠A = ∠B = ∠C, which is what needed to be proven.
Answer: Proven.
In today's lesson we looked at an isosceles triangle and studied its basic properties. In the next lesson we will solve problems on the topic of isosceles triangles, on calculating the area of an isosceles and equilateral triangle.
- Alexandrov A.D., Werner A.L., Ryzhik V.I. and others. Geometry 7. - M.: Education.
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B. and others. Geometry 7. 5th ed. - M.: Enlightenment.
- Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolova, ed. Sadovnichego V.A. - M.: Education, 2010.
- Dictionaries and encyclopedias on Academician ().
- Festival of pedagogical ideas “Open Lesson” ().
- Kaknauchit.ru ().
1. No. 29. Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolova, ed. Sadovnichego V.A. - M.: Education, 2010.
2. The perimeter of an isosceles triangle is 35 cm, and the base is three times smaller than the side. Find the sides of the triangle.
3. Given: AB = BC. Prove that ∠1 = ∠2.
4. The perimeter of an isosceles triangle is 20 cm, one of its sides is twice as large as the other. Find the sides of the triangle. How many solutions does the problem have?
Among all triangles, there are two special types: right triangles and isosceles triangles. Why are these types of triangles so special? Well, firstly, such triangles extremely often turn out to be the main characters in the problems of the Unified State Exam in the first part. And secondly, problems about right and isosceles triangles are much easier to solve than other geometry problems. You just need to know a few rules and properties. All the most interesting things are discussed in the corresponding topic, but now let’s look at isosceles triangles. And first of all, what is an isosceles triangle? Or, as mathematicians say, what is the definition of an isosceles triangle?
See what it looks like:
Like a right triangle, an isosceles triangle has special names for its sides. Two equal sides are called sides, and the third party - basis.
And again pay attention to the picture:
It could, of course, be like this:
So be careful: lateral side - one of two equal sides in an isosceles triangle, and the basis is a third party.
Why is an isosceles triangle so good? To understand this, let's draw the height to the base. Do you remember what height is?
What happened? From one isosceles triangle we get two rectangular ones.
This is already good, but this will happen in any, even the most “oblique” triangle.
How is the picture different for an isosceles triangle? Look again:
Well, firstly, of course, it is not enough for these strange mathematicians to just see - they must certainly prove. Otherwise, suddenly these triangles are slightly different, but we will consider them the same.
But don't worry: in this case, proving is almost as easy as seeing.
Shall we start? Look closely, we have:
And that means! Why? Yes, we will simply find and, and from the Pythagorean theorem (remembering at the same time that)
Are you sure? Well, now we have
And on three sides - the easiest (third) sign of equality of triangles.
Well, our isosceles triangle has divided into two identical rectangular ones.
See how interesting it is? It turned out that:
How do mathematicians usually talk about this? Let's go in order:
(Remember here that the median is a line drawn from a vertex that divides the side in half, and the bisector is the angle.)
Well, here we discussed what good things can be seen if given an isosceles triangle. We deduced that in an isosceles triangle the angles at the base are equal, and the height, bisector and median drawn to the base coincide.
And now another question arises: how to recognize an isosceles triangle? That is, as mathematicians say, what are signs of an isosceles triangle?
And it turns out that you just need to “turn” all the statements the other way around. This, of course, does not always happen, but an isosceles triangle is still a great thing! What happens after the “turnover”?
Well, look:
If the height and median coincide, then:
If the height and bisector coincide, then: 
If the bisector and the median coincide, then:
Well, don’t forget and use:
- If you are given an isosceles triangular triangle, feel free to draw the height, get two right triangles and solve the problem about a right triangle.
- If given that two angles are equal, then a triangle exactly isosceles and you can draw the height and….(The House That Jack Built…).
- If it turns out that the height is divided in half, then the triangle is isosceles with all the ensuing bonuses.
- If it turns out that the height divides the angle between the floors - it is also isosceles!
- If a bisector divides a side in half or a median divides an angle, then this also happens only in an isosceles triangle
Let's see what it looks like in tasks.
Problem 1(the simplest)
In a triangle, sides and are equal, a. Find.
We decide:
First the drawing.
What is the basis here? Certainly, .
Let's remember what if, then and.
Updated drawing:
Let's denote by. What is the sum of the angles of a triangle? ?
We use:
That's answer: .
Not difficult, right? I didn't even have to adjust the height.
Problem 2(Also not very tricky, but we need to repeat the topic)
In a triangle, . Find.
We decide:
The triangle is isosceles! We draw the height (this is the trick with which everything will be decided now).
Now let’s “cross out from life”, let’s just look at it.
So, we have:
Let's remember the tabular values of cosines (well, or look at the cheat sheet...)
All that remains is to find: .
Answer: .
Note that we here Very required knowledge regarding right triangles and “tabular” sines and cosines. Very often this happens: the topics , “Isosceles triangle” and in problems go together, but are not very friendly with other topics.
Isosceles triangle. Average level.
These two equal sides are called sides, A the third side is the base of an isosceles triangle.
Look at the picture: and - the sides, - the base of the isosceles triangle.
Let's use one picture to understand why this happens. Let's draw a height from a point.
This means that all corresponding elements are equal.
All! In one fell swoop (height) they proved all the statements at once.
And remember: to solve a problem about an isosceles triangle, it is often very useful to lower the height to the base of the isosceles triangle and divide it into two equal right triangles.
Signs of an isosceles triangle
The converse statements are also true:
Almost all of these statements can again be proven “in one fell swoop.”
1. So, let in turned out to be equal and.
Let's check the height. Then
2. a) Now let in some triangle height and bisector coincide.
2. b) And if the height and median coincide? Everything is almost the same, no more complicated!
 |
- on two sides |
2. c) But if there is no height, which is lowered to the base of an isosceles triangle, then there are no initially right triangles. Badly!
But there is a way out - read it in the next level of the theory, since the proof here is more complicated, but for now just remember that if the median and bisector coincide, then the triangle will also turn out to be isosceles, and the height will still coincide with these bisector and median.
Let's summarize:
- If the triangle is isosceles, then the angles at the base are equal, and the altitude, bisector and median drawn to the base coincide.
- If in some triangle there are two equal angles, or some two of the three lines (bisector, median, altitude) coincide, then such a triangle is isosceles.
Isosceles triangle. Brief description and basic formulas
An isosceles triangle is a triangle that has two equal sides.
Signs of an isosceles triangle:
- If in a certain triangle two angles are equal, then it is isosceles.
- If in some triangle they coincide:
A) height and bisector or
b) height and median or
V) median and bisector,
drawn to one side, then such a triangle is isosceles.
Well, the topic is over. If you are reading these lines, it means you are very cool.
Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!
Now the most important thing.
You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough...
For what?
For successfully passing the Unified State Exam, for entering college on a budget and, MOST IMPORTANTLY, for life.
I won’t convince you of anything, I’ll just say one thing...
People who have received a good education earn much more than those who have not received it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...
But think for yourself...
What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?
GAIN YOUR HAND BY SOLVING PROBLEMS ON THIS TOPIC.
You won't be asked for theory during the exam.
You will need solve problems against time.
And, if you haven’t solved them (A LOT!), you’ll definitely make a stupid mistake somewhere or simply won’t have time.
It's like in sports - you need to repeat it many times to win for sure.
Find the collection wherever you want, necessarily with solutions, detailed analysis and decide, decide, decide!
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Find problems and solve them!
The properties of an isosceles triangle are expressed by the following theorems.
Theorem 1. In an isosceles triangle, the angles at the base are equal.
Theorem 2. In an isosceles triangle, the bisector drawn to the base is the median and altitude.
Theorem 3. In an isosceles triangle, the median drawn to the base is the bisector and the altitude.
Theorem 4. In an isosceles triangle, the altitude drawn to the base is the bisector and the median.
Let us prove one of them, for example Theorem 2.5.
Proof. Let us consider an isosceles triangle ABC with base BC and prove that ∠ B = ∠ C. Let AD be the bisector of triangle ABC (Fig. 1). Triangles ABD and ACD are equal according to the first sign of equality of triangles (AB = AC by condition, AD is a common side, ∠ 1 = ∠ 2, since AD is a bisector). From the equality of these triangles it follows that ∠ B = ∠ C. The theorem is proven.
Using Theorem 1, the following theorem is established.
Theorem 5. The third criterion for the equality of triangles. If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent (Fig. 2).
Comment. The sentences established in examples 1 and 2 express the properties of the perpendicular bisector of a segment. From these proposals it follows that perpendicular bisectors to the sides of a triangle intersect at one point.
Example 1. Prove that a point in the plane equidistant from the ends of a segment lies on the perpendicular bisector to this segment.
Solution. Let point M be equidistant from the ends of segment AB (Fig. 3), i.e. AM = BM.
Then Δ AMV is isosceles. Let us draw a straight line p through the point M and the midpoint O of the segment AB. By construction, the segment MO is the median of the isosceles triangle AMB, and therefore (Theorem 3), and the height, i.e., the straight line MO, is the perpendicular bisector to the segment AB.
Example 2. Prove that each point of the perpendicular bisector to a segment is equidistant from its ends.
Solution. Let p be the perpendicular bisector to segment AB and point O be the midpoint of segment AB (see Fig. 3).
Consider an arbitrary point M lying on the straight line p. Let's draw segments AM and BM. Triangles AOM and BOM are equal, since their angles at vertex O are right, leg OM is common, and leg OA is equal to leg OB by condition. From the equality of triangles AOM and BOM it follows that AM = BM.
Example 3. In triangle ABC (see Fig. 4) AB = 10 cm, BC = 9 cm, AC = 7 cm; in triangle DEF DE = 7 cm, EF = 10 cm, FD = 9 cm.
Compare triangles ABC and DEF. Find the corresponding equal angles.
Solution. These triangles are equal according to the third criterion. Correspondingly, equal angles: A and E (lie opposite equal sides BC and FD), B and F (lie opposite equal sides AC and DE), C and D (lie opposite equal sides AB and EF).
Example 4. In Figure 5, AB = DC, BC = AD, ∠B = 100°.
Find angle D.
Solution. Consider triangles ABC and ADC. They are equal according to the third criterion (AB = DC, BC = AD by condition and side AC is common). From the equality of these triangles it follows that ∠ B = ∠ D, but angle B is equal to 100°, which means angle D is equal to 100°.
Example 5. In an isosceles triangle ABC with base AC, the exterior angle at vertex C is 123°. Find the size of angle ABC. Give your answer in degrees.
Video solution.
Geometry is not just a subject in school in which you need to get an excellent grade. This is also knowledge that is often required in life. For example, when building a house with a high roof, it is necessary to calculate the thickness of the logs and their number. This is not difficult if you know how to find the height in an isosceles triangle. Architectural structures are based on knowledge of the properties of geometric shapes. The shapes of buildings often visually resemble them. Egyptian pyramids, milk bags, artistic embroidery, northern paintings and even pies - these are all triangles surrounding a person. As Plato said, the whole world is based on triangles.
Isosceles triangle
A triangle is isosceles if it has two equal sides. They are always called side. The side whose dimensions differ is called the base.
Basic Concepts
Like any science, geometry has its basic rules and concepts. There are quite a lot of them. Let us consider only those without which our topic will be somewhat incomprehensible.
Height is a straight line drawn perpendicular to the opposite side.
The median is a segment directed from any vertex of a triangle exclusively to the middle of the opposite side.
An angle bisector is a ray that bisects the angle.
The bisector of a triangle is a straight line, or rather, a segment connecting a vertex to the opposite side.
It is very important to remember that the bisector of an angle is necessarily a ray, and the bisector of a triangle is part of such a ray.
Angles at the base
The theorem states that the angles at the base of any isosceles triangle are always equal. Proving this theorem is very simple. Consider the illustrated isosceles triangle ABC, in which AB = BC. From angle ABC it is necessary to draw a bisector VD. Now we should consider the two resulting triangles. According to the condition AB = BC, the side WD of the triangles is common, and the angles AVD and SVD are equal, because WD is a bisector. Remembering the first sign of equality, we can safely conclude that the triangles in question are equal. Therefore, all corresponding angles are equal. And, of course, the sides, but we’ll return to this point later.
Height of an isosceles triangle
The main theorem on which the solution to almost all problems is based is as follows: the height in an isosceles triangle is the bisector and the median. To understand its practical meaning (or essence), you should make an auxiliary manual. To do this, you need to cut out an isosceles triangle from paper. The easiest way to do this is from a regular notebook sheet in a box.
Bend the resulting triangle in half, aligning the sides. What happened? Two equal triangles. Now you should check your guesses. Unfold the resulting origami. Draw a fold line. Using a protractor, check the angle between the drawn line and the base of the triangle. What does a 90 degree angle mean? That the drawn line is perpendicular. By definition - height. We figured out how to find the height in an isosceles triangle. Now let's deal with the vertex angles. Using the same protractor, check the angles formed by the now height. They are equal. This means that the height is also a bisector. Armed with a ruler, measure the segments into which the height of the base is divided. They are equal. Therefore, the altitude in an isosceles triangle bisects the base and is the median.
Proof of the theorem
A visual aid clearly demonstrates the truth of the theorem. But geometry is a fairly accurate science, so it requires proof.
While considering the equality of angles at the base, the equality of triangles was proven. Recall that WD is a bisector, and triangles AVD and SVD are equal. The conclusion was this: the corresponding sides of the triangle and, naturally, the angles are equal. So, BP = DM. Therefore, VD is the median. It remains to prove that VD is a height. Based on the equality of the triangles under consideration, it turns out that angle ADV is equal to angle DDV. But these two angles are adjacent, and, as you know, add up to 180 degrees. Therefore, what are they equal to? Of course, 90 degrees. Thus, VD is the height in an isosceles triangle drawn to the base. Q.E.D.
Main features
- To successfully solve problems, you should remember the main features of isosceles triangles. They are, as it were, the inverse of the theorems.
- If, while solving a problem, two angles are found to be equal, then you are dealing with an isosceles triangle.
- If you can prove that the median is also the altitude of the triangle, feel free to conclude that the triangle is isosceles.
- If the bisector is also a height, then, based on the main characteristics, the triangle is classified as isosceles.
- And, of course, if the median also acts as a height, then such a triangle is isosceles.
Height Formula 1
However, most problems require finding the arithmetic value of height. That is why we will consider how to find the height in an isosceles triangle.
Let's return to the figure ABC presented above, in which a is the sides, b is the base. VD is the height of this triangle, it is designated h.
What is the AED triangle? Since VD is the height, the triangle ABC is a right triangle, the leg of which must be found. Using the Pythagorean formula, we get:
AB² = AD² + VD²
Determining the VD from the expression and substituting the previously accepted notations, we obtain:
Н² = а² - (в/2)².
You need to extract the root:
Н = √а² - в²/4.
If you remove ¼ from under the root sign, the formula will look like:
H = ½ √4a² - b².
This is how you find the height in an isosceles triangle. The formula follows from the Pythagorean theorem. Even if you forget this symbolic notation, then, knowing the method of finding, you can always derive it.
Height Formula 2
The formula described above is the basic one and is most often used when solving most geometric problems. But she's not the only one. Sometimes the condition, instead of the base, gives the value of the angle. Given such data, how to find the height in an isosceles triangle? To solve such problems, it is advisable to use another formula:
where H is the height directed towards the base,
a - side side,
α - angle at the base.
If the problem is given the value of the vertex angle, then the height in an isosceles triangle is found as follows:
Н = а/cos (β/2),
where H is the height lowered to the base,
β - vertex angle,
a - side.
Right isosceles triangle
A triangle whose vertex is 90 degrees has a very interesting property. Consider ABC. As in previous cases, HP is the height directed towards the base.
The angles at the base are equal. It won’t be difficult to calculate them:
α = (180 - 90)/2.
Thus, the angles at the base are always 45 degrees. Now consider triangle ADV. It is also rectangular. Let's find the angle AVD. By simple calculations we get 45 degrees. And, therefore, this triangle is not only right-angled, but also isosceles. Sides AD and HP are lateral sides and are equal to each other.
But side AD is at the same time half of side AC. It turns out that the height in an isosceles triangle is equal to half the base, and if we write it in the form of a formula, we get the following expression:
It should be remembered that this formula is an exclusively special case and can only be used for right isosceles triangles.
Golden triangles
The golden triangle is very interesting. In this figure, the ratio of the side to the base is equal to a value called the Phidias number. The angle located at the top is 36 degrees, at the base - 72 degrees. The Pythagoreans admired this triangle. The principles of the Golden Triangle form the basis of many immortal masterpieces. The well-known one is built on the intersection of isosceles triangles. Leonardo da Vinci used the principle of the “golden triangle” for many of his creations. The composition of “La Gioconda” is based precisely on the figures that create a regular star-shaped pentagon.
The painting “Cubism”, one of the creations of Pablo Picasso, captivates the eye with its isosceles triangles as its basis.