What is the third angle in an isosceles triangle. Isosceles triangle. Detailed theory with examples. Radius of an inscribed circle in an isosceles triangle
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 very often turn out to be the main actors in the tasks of the Unified State Examination of the first part. And secondly, problems about right-angled and isosceles triangles are much easier to solve than other problems in geometry. You just need to know a few rules and properties. All the most interesting is discussed in the corresponding topic, and now we will consider 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, look at 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, two right-angled ones turned out.
This is already good, but this will happen in any, the most “oblique” triangle.
What is the difference between the picture for an isosceles triangle? Look again:
Well, firstly, of course, it is not enough for these strange mathematicians to simply see - they must certainly prove. And then suddenly these triangles are slightly different, and we will consider them the same.
But don't worry: this case proving is almost as easy as seeing.
Shall we start? Look carefully, we have:
And, therefore,! Why? Yes, we just 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 the equality of triangles.
Well, our isosceles triangle is divided into two identical rectangular ones.
See how interesting? It turned out that:
How is it customary for mathematicians to talk about this? Let's go in order:
(We recall here that the median is a line drawn from the vertex that bisects the side, and the bisector is the angle.)
Well, here we discussed what good can be seen if given an isosceles triangle. We have deduced that in an isosceles triangle the angles at the base are equal, and the height, bisector and median drawn to the base are the same.
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 on the contrary. This, of course, does not always happen, but an isosceles triangle is still a great thing! What happens after the "reversal"?
Well look here:
If height and median are the same, then:
If the height and bisector are the same, then: 
If the bisector and median are the same, then:
Well, do not forget and use:
- If an isosceles triangular triangle is given, feel free to draw a height, get two right triangles and solve the problem already about a right triangle.
- If given that two angles are equal, then the triangle exactly isosceles and you can draw a height and .... (The house that Jack built ...).
- If it turned out that the height is divided in half by the side, then the triangle is isosceles with all the ensuing bonuses.
- If it turned out that the height divided the angle to the floors - also isosceles!
- If the bisector divided the side in half or the median - the angle, then this also happens only in an isosceles triangle
Let's see how it looks in tasks.
Task 1(the simplest)
In a triangle, the sides and are equal, a. Find.
We decide:
First a drawing.
What is the basis here? Of course, .
We recall that if, then and.
Updated drawing:
Let's designate for. What is the sum of the angles of the triangle? ?
We use:
That's answer: .
Easy, right? I didn't even have to go high.
Task 2(Also not very tricky, but you need to repeat the theme)
In a triangle, Find.
We decide:
The triangle is isosceles! We draw the height (this is the focus, with the help of which everything will be decided now).
Now "we delete from life", we will consider only.
So, in we have:
We recall the tabular values of cosines (well, or look at the cheat sheet ...)
It remains to find: .
Answer: .
Note that we are here very required knowledge regarding the right triangle and the "tabular" sines and cosines. Very often this happens: the topics, “Isosceles Triangle” and in puzzles go in bundles, but they are not very friendly with other topics.
Isosceles triangle. Average level.
These two equal sides called sides, a the third side is the base of an isosceles triangle.
Look at the picture: and - sides, is the base of an isosceles triangle.
Let's see in one picture why this is so. Draw a height from a point.
This means that all corresponding elements are equal.
Everything! In one fell swoop (height) all the statements were proved at once.
And you remember: to solve the isosceles triangle problem, 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 proved "in one fell swoop".
1. So, let v turned out to be equal and.
Let's take the height. Then
2. a) Now let in some triangle same height and bisector.
2. b) And if the height and median are the same? Everything is almost the same, nothing more complicated!
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- on two legs |
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 theory, because the proof is more complicated here, but for now just remember that if the median and the bisector coincide, then the triangle will also be isosceles, and the height will still coincide with these bisector and median.
To summarize:
- If the triangle is isosceles, then the angles at the base are equal, and the height, bisector and median drawn to the base are the same.
- If in some triangle there are two equal angles, or some two of the three lines (bisector, median, height) 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 a triangle has two equal angles, then it is isosceles.
- If in some triangle coincide:
a) height and bisector or
b) height and median or
in) median and bisector,
drawn to one side, then such a triangle is isosceles.
Well, the topic is over. If you are reading these lines, then you are very cool.
Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!
Now the most important thing.
You've figured out the theory on this topic. And, I repeat, it's ... it's 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 successful passing the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.
I will not convince you of anything, I will 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 much 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 exam and be ultimately ... happier?
FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.
On the exam, you will not be asked theory.
You will need solve problems on time.
And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.
It's like in sports - you need to repeat many times to win for sure.
Find a collection anywhere you want necessarily with solutions, detailed analysis and decide, decide, decide!
You can use our tasks (not necessary) and we certainly recommend them.
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In conclusion...
If you don't like our tasks, find others. Just don't stop with theory.
“Understood” and “I know how to solve” are completely different skills. You need both.
Find problems and solve!
Geometry is not only a subject in school in which you need to get an excellent mark. It 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 easy if you know how to find the height in an isosceles triangle. Architectural structures are based on knowledge of properties 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 that surround 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 lateral. The side whose dimensions differ is called the base.
Basic concepts
Like any science, geometry has its own basic rules and concepts. There are enough of them. Let us consider only those without which our topic would be somewhat incomprehensible.
The height is a straight line drawn perpendicular to the opposite side.
A median is a segment directed from any vertex of a triangle exclusively to the midpoint of the opposite side.
An angle bisector is a ray that bisects an angle.
The bisector of a triangle is a straight line, or rather, a segment connecting the vertex with 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 a part of such a ray.
Base angles
The theorem states that the angles located at the base of any isosceles triangle are always equal. It is very easy to prove this theorem. Consider the depicted isosceles triangle ABC, where AB=BC. From the angle ABC it is necessary to draw the bisector of the VD. Now we should consider the two resulting triangles. By the condition AB \u003d BC, the side of the VD is common for the triangles, and the angles ABD and SVD are equal, because the VD is a bisector. Remembering the first sign of equality, we can safely conclude that the triangles under consideration are equal. Therefore, all corresponding angles are equal. And, of course, the parties, but we will return to this point later.
Height of an isosceles triangle
The main theorem, on which the solution of almost all problems is based, sounds like this: the height in an isosceles triangle is a bisector and a median. To understand its practical meaning (or essence), an auxiliary manual should be made. To do this, cut out an isosceles triangle from paper. The easiest way to do this is from an ordinary notebook sheet in a box.
Bend the resulting triangle in half, aligning the sides. What happened? Two equal triangles. Now we need to check our guesses. Expand 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 a perpendicular. By definition, height. How to find the height in an isosceles triangle, we figured it out. Now let's deal with the corners at the top. Using the same protractor, check the angles formed now by the height. They are equal. This means that the height is also a bisector. Armed with a ruler, measure the segments into which the base divides the height. They are equal. Therefore, the height in an isosceles triangle bisects the base and is the median.
Proof of the theorem
The visual aid vividly demonstrates the truth of the theorem. But geometry is a rather exact science, therefore it requires proof.
During the consideration of the equality of angles at the base, the equality of triangles was proved. Recall that BD is a bisector, and triangles ABD and SVD are congruent. The conclusion was that the corresponding sides of the triangle and, of course, the angles are equal. So BP = SD. Therefore, PD is the median. It remains to prove that VD is a height. Based on the equality of the considered triangles, it turns out that the angle ADV is equal to the angle SDV. But these two angles are adjacent and are known to 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 seem to be the opposite of the theorems.
- If in the course of solving the problem the equality of two angles is found, then you are dealing with an isosceles triangle.
- If you managed to prove that the median is also the height of the triangle, feel free to conclude that the triangle is isosceles.
- If the bisector is also a height, then, based on the main features, 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, for most tasks it is required to find the arithmetic value of the height. That is why we will consider how to find the height in an isosceles triangle.
Let's return to the ABC figure presented above, in which a is the sides, b is the base. VD - the height of this triangle, it has the designation h.
What is the triangle ABD? Since VD is the height, then the triangle ABD is a rectangular one, the leg of which must be found. Using the Pythagorean formula, we get:
AB² = AD² + VD²
Having determined the VD from the expression and substituting the previously accepted designations, we obtain:
H² \u003d a² - (b / 2)².
Extract the root:
H \u003d √a² - v² / 4.
If we take out from under the root sign ¼, then the formula will look like:
H \u003d ½ √4a² - in².
This is the altitude in an isosceles triangle. The formula follows from the Pythagorean theorem. Even if you forget this symbolic record, then, knowing the method of finding, you can always deduce it.
Height formula 2
The formula described above is the main one and is most often used in solving most geometric problems. But she's not the only one. Sometimes in the condition, instead of the base, the value of the angle is given. With 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
α - angle at the base.
If the value of the angle at the vertex is given in the problem, then the height in an isosceles triangle is found as follows:
H \u003d a / cos (β / 2),
where H is the height lowered to the base,
β - vertex angle,
a - side.
right isosceles triangle
A very interesting property has a triangle whose apex is 90 degrees. Consider ABC. As in previous cases, VD is the height directed towards the base.
The base angles are equal. It is not difficult to calculate them:
α = (180 - 90)/2.
Thus, the angles at the base are always 45 degrees. Now consider the triangle ADV. It is also rectangular. Let's find the angle ABD. By simple calculations we get 45 degrees. And, therefore, this triangle is not only right-angled, but also isosceles. The sides AD and VD 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 written as a formula, we get the following expression:
It should not be forgotten that this formula is an exceptionally special case, and can only be used for right-angled isosceles triangles.
golden triangles
The golden triangle is very interesting. In this figure, the ratio of the side to the base is equal to the value called the number of Phidias. The angle located at the top is 36 degrees, at the base - 72 degrees. This triangle was admired by the Pythagoreans. The principles of the golden triangle form the basis of many immortal masterpieces. Known to everyone, it is built at the intersection of isosceles triangles. For many creations, Leonardo da Vinci used the principle of the "golden triangle". The composition of the Gioconda is based precisely on the figures that create a regular star pentagon.
The painting "Cubism", one of the creations of Pablo Picasso, captivates the eye with isosceles triangles.
Isosceles triangle is a triangle in which the lengths of its two sides are equal to each other.Note. From the definition of an isosceles triangle, it follows that a regular triangle is also an isosceles triangle. However, it must be remembered that the converse is not true.
Properties of an isosceles triangle
The properties below are used in solving problems. Since they are widely known, it is understood that they need no explanation. Therefore, in the texts of the tasks, the reference to them is omitted.- corners equal between themselves.
- Bisectors, medians and heights drawn from the angles opposite the equal sides of the triangle, equal between themselves.
- Bisector, median and height, drawn to the base, match between themselves.
- Centers of inscribed and circumscribed circles lie on the height, bisector and median (they coincide) drawn to the base.
- corners opposite the equal sides of an isosceles triangle, always sharp.
The sides in an isosceles triangle can be calculated using formulas expressing their length in terms of other sides and angles, the magnitude of which is known.
The lateral side of an isosceles triangle is equal to the quotient of dividing the base by the double cosine of the angle at the base (Formula 1). This identity can be obtained by simple transformations from the cosine theorem.
The base of an isosceles triangle is equal to the product of the side by Square root from twice the difference of unity and the cosine of the angle at the vertex (Formula 2)
The base of an isosceles triangle is equal to twice the product of the lateral side and the sine of half the angle at the apex. (Formula 3)
The base of an isosceles triangle is equal to twice the product of the lateral side and the cosine of the angle at its base (Formula 4).
Radius of an inscribed circle in an isosceles triangle
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The symbols in the formulas can be seen in the figure above. The radius of the inscribed circle for an isosceles triangle can be found based on the values of the base and each side. (Formula 1) The radius of the inscribed circle for an isosceles triangle can be determined based on the values of the base and the height drawn to this base (Formula 2) The radius of a circle inscribed in an isosceles triangle can also be calculated through the length of the lateral side and the height drawn to the base of the triangle (Formula 3) Knowing the angle between the sides and the length of the base also allows you to determine the radius of the inscribed circle (Formula 4) A similar formula (5) allows you to determine the radius of the inscribed circle through the sides and the angle between them |
Signs of an isosceles triangle
A triangle that has the following features is isosceles.- Two angles of a triangle are equal
- The height is the same as the median
- Height coincides with the bisector
- The bisector is the same as the median
- Two heights are equal
- The two medians are equal
- Two bisectors are equal
Area of an isosceles triangle
The area of an isosceles triangle is found by the following formulas:
, where
a- the length of one of the two equal sides of the triangle
b- base length
α - the value of one of the two equal angles at the base
β - the angle between the equal sides of the triangle and the opposite of its base.
A triangle with two equal sides is called an isosceles triangle. These sides are called the sides, and the third side is called the base. In this article, we will tell you about the properties of an isosceles triangle.
Theorem 1
The angles near the base of an isosceles triangle are equal to each other
Proof of the theorem.
Suppose we have an isosceles triangle ABC whose base is AB. Let's look at triangle BAC. These triangles, by the first sign, are equal to each other. So it is, because BC = AC, AC = BC, angle ACB = angle ACB. It follows from this that angle BAC = angle ABC, because these are the corresponding angles of our triangles equal to each other. Here is the property of the angles of an isosceles triangle.
Theorem 2
The median in an isosceles triangle drawn to its base is also the height and bisector
Proof of the theorem.
Let's say we have an isosceles triangle ABC whose base is AB and CD is the median we drew to its base. In triangles ACD and BCD, angle CAD = angle CBD, as the corresponding angles at the base of an isosceles triangle (Theorem 1). And side AC = side BC (by definition of an isosceles triangle). Side AD \u003d side BD, After all, point D divides segment AB into equal parts. Hence it follows that triangle ACD = triangle BCD.
From the equality of these triangles, we have the equality of the corresponding angles. That is, angle ACD = angle BCD and angle ADC = angle BDC. Equation 1 implies that CD is a bisector. And angle ADC and angle BDC are adjacent angles, and from equality 2 it follows that they are both right angles. It turns out that CD is the height of the triangle. This is the property of the median of an isosceles triangle.
And now a little about the signs of an isosceles triangle.
Theorem 3
If two angles in a triangle are congruent, then the triangle is isosceles.
Proof of the theorem.
Let's say we have a triangle ABC in which angle CAB = angle CBA. Triangle ABC = triangle BAC by the second criterion of equality between triangles. So it is, because AB = BA; angle CBA = angle CAB, angle CAB = angle CBA. From such an equality of triangles, we have the equality of the corresponding sides of the triangle - AC = BC. Then it turns out that triangle ABC is isosceles.
Theorem 4
If in any triangle its median is also its height, then such a triangle is isosceles
Proof of the theorem.
In the triangle ABC we draw the median CD. It will also be height. Right triangle ACD = right triangle BCD, since leg CD is common to them, and leg AD = leg BD. From this it follows that their hypotenuses are equal to each other, as the corresponding parts of equal triangles. This means that AB = BC.
Theorem 5
If three sides of a triangle are equal to three sides of another triangle, then these triangles are congruent
Proof of the theorem.
Suppose we have a triangle ABC and a triangle A1B1C1 such that the sides are AB = A1B1, AC = A1C1, BC = B1C1. Consider the proof of this theorem by contradiction.
Assume that these triangles are not equal to each other. Hence we have that the angle BAC is not equal to the angle B1A1C1, the angle ABC is not equal to the angle A1B1C1, the angle ACB is not equal to the angle A1C1B1 at the same time. Otherwise, these triangles would be equal according to the above criterion.
Assume that triangle A1B1C2 = triangle ABC. The vertex C2 of a triangle lies with the vertex C1 relative to the line A1B1 in the same half-plane. We assumed that the vertices C2 and C1 do not coincide. Assume that point D is the midpoint of segment C1C2. So we have isosceles triangles B1C1C2 and A1C1C2, which have a common base C1C2. It turns out that their medians B1D and A1D are also their heights. This means that line B1D and line A1D are perpendicular to line C1C2.
B1D and A1D have different points B1 and A1 and therefore cannot coincide. But after all, through the point D of the line C1C2 we can draw only one line perpendicular to it. We've got a contradiction.
Now you know what are the properties of an isosceles triangle!
