Bisector of a triangle. Detailed theory with examples (2019)
You will need
- - right triangle;
- - known length of legs;
- - known length of the hypotenuse;
- - known angles and one of the sides;
- - the known lengths of the parts into which the bisector divides the hypotenuse.
Instructions
Use the following theorem: the relations of the legs and the relations of adjacent segments on which there is a direct angle divides the hypotenuse are equal. That is, divide the legs into each other and equate them to the ratio x/(c-x). At the same time, make sure that the numerator contains the leg adjacent to x. Solve the resulting equation and find x.
Having found out the length of the segments for which the bisector of a straight line angle divided the hypotenuse, find the length of the hypotenuse itself using the theorem of sines. You know the angle between the leg and the bisector - 45⁰, the two sides of the internal triangle too.
Substitute the data into the sine theorem: x/sin45⁰=l/sinα. Simplifying the expression, you get l=2xsinα/√2. Substitute the found x: l=2c*cosα*sinα/√2(sinα+cosα)=c*sin2α/2cos(45⁰-α). This is the bisector of the line angle, expressed through the hypotenuse.
If you are given legs, you have two options: either find the length of the hypotenuse using the Pythagorean theorem, according to which the sum of the squares of the legs is equal to the square of the hypotenuse and solve in the above manner. Or use the following ready-made formula: l=√2*ab/(a+b), where a and b are the lengths of the legs.
Sources:
- how to find the length of a straight line
Dividing an angle in half and calculating the length of a line drawn from its top to the opposite side is something that cutters, surveyors, installers and people of some other professions need to be able to do.
You will need
- Tools Pencil Ruler Protractor Sine and Cosine Tables Mathematical formulas and concepts: Definition of a bisector Theorems of sines and cosines Bisector theorem
Instructions
Construct a triangle of the required size, depending on what is given to you? dfe sides and the angle between them, three sides or two angles and the side located between them.
Label the vertices of the corners and sides with the traditional Latin letters A, B and C. The vertices of the corners are denoted by , and the opposite sides are denoted by lowercase letters. Label the angles with Greek letters?,? And?
Using the theorems of sines and cosines, calculate the angles and sides triangle.
Remember bisectors. Bisector - dividing an angle in half. Angle bisector triangle divides the opposite into two segments, which are equal to the ratio of the two adjacent sides triangle.
Draw the bisectors of the angles. Label the resulting segments with the names of the angles, written in lowercase letters, with a subscript l. Side c is divided into segments a and b with indices l.
Calculate the lengths of the resulting segments using the law of sines.
Video on the topic
note
The length of the segment, which is simultaneously the side of the triangle formed by one of the sides of the original triangle, the bisector and the segment itself, is calculated using the law of sines. In order to calculate the length of another segment of the same side, use the ratio of the resulting segments and the adjacent sides of the original triangle.
To avoid confusion, draw bisectors of different angles different colors.
Tip 3: How to find the bisector in right triangle
A bisector is a ray that divides an angle in half. The bisector, in addition to this, has many more properties and functions. And in order to calculate its length in rectangular triangle, you will need the formulas and instructions below.
You will need
- - calculator
Instructions
Multiply side a, side b, the semi-perimeter of the triangle p and the number four 4*a*b. Next, the resulting amount must be multiplied by the difference between the half-perimeter p and side c 4*a*b*(p-c). Extract the root of what you got earlier. SQR(4*a*b*(p-c)). And divide the result by the sum of sides a and b. Thus, we have obtained one of the formulas for finding the bisector using Stewart's theorem. It can be interpreted in a different way, presenting it this way: SQR(a*b*(a+b+c)(a+b-c)). There are several more options for this formula, obtained on the basis of the same theorem.
Multiply side a by side b. From the result, subtract the lengths of the segments e and d into which the bisector l divides side c. The results look like this: a*b-e*d. Next, you need to extract the root of the presented difference SQR (a*b-e*d). This is another method for the length of the bisector in triangles. Do all calculations carefully, repeating at least 2 times for possible errors.
Multiply two by sides a and b, plus the cosine of angle c divided in half. Next, the resulting product must be divided by the sum of sides a and b. Provided the cosines are known, this method of calculation will be the most convenient for you.
Subtract the cosine of angle b from the cosine of angle a. Then divide the resulting difference in half. The divisor that we will need later has been calculated. Now all that remains is to divide the height drawn to side c by the previously calculated number. Now another calculation method has been demonstrated for finding the bisector in a rectangular triangle. The choice of method for finding the numbers you need is up to you, and also depends on what is provided in the conditions for this or that geometric figure.
Video on the topic
Let two intersecting lines given by their equations be given. It is required to find the equation of a line that, passing through the point of intersection of these two lines, would exactly bisect the angle between them, that is, would be a bisector.
One of the basics of geometry is finding the bisector, the ray that bisects an angle. The bisector of a triangle is the part of the bisector of any angle. This is a segment from the vertex of the angle to the intersection with the opposite side of the triangle.
If you draw bisectors from all angles, they will intersect at one point, which is called the center of the inscribed triangle.
You can calculate the bisector if you know the length of the side that it bisects, or the size of the angles of the triangle.
Bisector of an isosceles triangle
Since in an isosceles triangle two sides are equal to each other, then the bisectors of adjacent angles will be equal. Because The angles of the triangle are also equal.
When drawing a bisector from one of the corners, it will be considered the height of the given triangle and its median.
Problems of how to find the bisector of a triangle are solved using formulas.
To solve these formulas, the conditions must indicate the values of the lengths of the sides, or the values of the angles of the triangle. Knowing them, you can calculate the bisector using cosines or perimeter.
For example, take an isosceles triangle ABC and draw the bisector AE to the base BC. The resulting triangle AEB is right-angled. The bisector is its altitude, side AB is the hypotenuse of the right triangle, and BE and AE are the legs.
The Pythagorean theorem is applied - the square of the hypotenuse is equal to the sum of the squares of the legs. Based on it BE = v (AB - AE). Since AE is the median of triangle ABC, then side BE = BC/2. Thus BE = v(AB - (BC/4)).
If the base angle ABC is given, then the bisector of the triangle is AEB, AE = AB/sin(ABC). Base angle AEB, BAE = BAC/2. Therefore, the bisector AE = AB/cos (BAC/2).
How to find the bisector of a triangle inscribed in another triangle?
In an isosceles triangle ABC, draw side BC to side AC. This segment will be neither the bisector of the triangle nor its median. The Stewart formula applies here.
It is used to calculate the perimeter of a triangle - the sum of the lengths of all its sides. For ABC we calculate the semi-perimeter. This is the perimeter of the triangle divided in half.
P = (AB+ BC+ AC)/2. Using this formula, we calculate the bisector drawn to the side. VK = v(4*VS*AS*P (R-AV)/ (VS+AS).
By Stewart's theorem, you can also see that the bisector drawn to the other side of the triangle will be equal to VC, because these two sides of the triangle are equal to each other.
Bisector of a right triangle
In order to know how to find the bisector in a right triangle, you also need to use formulas. Do not forget that in a right triangle one angle is necessarily right, i.e. equal to 90 degrees. Thus, if the bisector starts from right angle, even if the condition does not indicate the sine or cosine of the angle, you can recognize them by the size of the angle.
- The bisector is found using Stewart's formula. If there is a triangle ABC, and its semi-perimeter is calculated as P = (AB+ BC+ AK)/2. Based on this, we calculate the bisector AE = v(4*VK*AK*P (P-AB)/ (VK+AK).
- The length of the bisector is determined in this way. AE = v (BK*AK) – (EB*EK), where EB and EK are the segments into which the bisector AE divides the side BK.
- Or you can use the cosines of the angles of a right triangle, if they are known. The bisector will be equal to (2*аb*(cos c/2))/(a+b).
- Or find the bisector like this. Using the formula (cos a) – (cos b)/2, find the divisor you need in the future. Next, the height drawn to side c is divided by the resulting value. To obtain cosines, you need to know the magnitude of the angles. Or calculate them based on the size of the only known angle - a right angle, 90 degrees.
Equilateral triangle
In such a triangle, all sides are equal to each other, and so are the angles. Therefore, all bisectors and medians will also be equal. If some of the side values are unknown, then the value of one side will be needed. Because the sides are equal. And the sizes of the angles too. Therefore, to find the bisector using the cosine formula, you need to know or calculate the value of only one of the angles.
The length of the median and bisector of a triangle is equal to - L.
The sides of the triangle are equal - a.
In triangle ABC, bisector AE = (ABCv3)/2.
The same formula is used to calculate the height and median of an equilateral triangle.
Scalene triangle
In such a triangle, all sides have different meanings, therefore the bisectors are not equal to each other.
Take a triangle with arbitrary side values. If some values of the sides are unknown, then they are calculated using the formula for the perimeter of a triangle.
After the angle bisectors have been drawn, it is worth adding a subscript1 to their designations. The segments into which the bisector divides the opposite side are also designated with the subscript 1.
The lengths of these segments are calculated using the sine theorem.
The length of the bisector is calculated as L = v ab – a1b1, where ab are the sides adjacent to the segments, and a1b1 is the product of the segments. The formula applies to all sides of a scalene triangle. The main thing is to know the lengths of the sides, or calculate them, knowing the values of the adjacent angles.
Average level
Bisector of a triangle. Detailed theory with examples (2019)
Bisector of a triangle and its properties
Do you know what the midpoint of a segment is? Of course you do. What about the center of the circle? Same. What is the midpoint of an angle? You can say that this doesn't happen. But why can a segment be divided in half, but an angle cannot? It’s quite possible - just not a dot, but…. line.
Do you remember the joke: a bisector is a rat that runs around the corners and divides the corner in half. So, the real definition of a bisector is very similar to this joke:
Bisector of a triangle- this is the bisector segment of an angle of a triangle connecting the vertex of this angle with a point on the opposite side.
Once upon a time, ancient astronomers and mathematicians discovered many interesting properties of the bisector. This knowledge has greatly simplified people's lives. It has become easier to build, count distances, even adjust the firing of cannons... Knowledge of these properties will help us solve some GIA and Unified State Examination tasks!
The first knowledge that will help with this is bisector of an isosceles triangle.
By the way, do you remember all these terms? Do you remember how they differ from each other? No? Not scary. Let's figure it out now.
So, base of an isosceles triangle- this is the side that is not equal to any other. Look at the picture, which side do you think this is? That's right - this is the side.
The median is a line drawn from the vertex of a triangle and dividing the opposite side (that's it again) in half.
Notice we don't say, "Median of an isosceles triangle." Do you know why? Because a median drawn from a vertex of a triangle bisects the opposite side in ANY triangle.
Well, the height is a line drawn from the top and perpendicular to the base. You noticed? We are again talking about any triangle, not just an isosceles one. The height in ANY triangle is always perpendicular to the base.
So, have you figured it out? Almost. To understand even better and forever remember what a bisector, median and height are, you need to compare them with each other and understand how they are similar and how they differ from each other. At the same time, in order to remember better, it is better to describe everything in “human language”. Then you will easily operate in the language of mathematics, but at first you do not understand this language and you need to comprehend everything in your own language.
So, how are they similar? The bisector, the median and the altitude - they all “come out” from the vertex of the triangle and rest on the opposite side and “do something” either with the angle from which they come out, or with the opposite side. I think it's simple, no?
How are they different?
- The bisector divides the angle from which it emerges in half.
- The median divides the opposite side in half.
- The height is always perpendicular to the opposite side.
That's it. It's easy to understand. And once you understand, you can remember.
Now the next question. Why in the case of isosceles triangle Is the bisector both the median and the height?
You can simply look at the figure and make sure that the median divides into two absolutely equal triangles. That's all! But mathematicians do not like to believe their eyes. They need to prove everything. Scary word? Nothing like that - it's simple! Look: both have equal sides and, they generally have a common side and. (- bisector!) And so it turns out that two triangles have two equal sides and an angle between them. We recall the first sign of equality of triangles (if you don’t remember, look in the topic) and conclude that, and therefore = and.
This is already good - it means it turned out to be the median.
But what is it?
Let's look at the picture - . And we got it. So, too! Finally, hurray! And.
Did you find this proof a bit heavy? Look at the picture - two identical triangles speak for themselves.
In any case, remember firmly:
Now it’s more difficult: we’ll count angle between bisectors in any triangle! Don't be afraid, it's not that tricky. Look at the picture:
Let's count it. Do you remember that the sum of the angles of a triangle is?
Let's apply this amazing fact.
On the one hand, from:
That is.
Now let's look at:
But bisectors, bisectors!
Let's remember about:
Now through the letters
\angle AOC=90()^\circ +\frac(\angle B)(2)
Isn't it surprising? It turned out that the angle between the bisectors of two angles depends only on the third angle!
Well, we looked at two bisectors. What if there are three of them??!! Will they all intersect at one point?
Or will it be like this?
How do you think? So mathematicians thought and thought and proved:
Isn't that great?
Do you want to know why this happens?
So...two right triangles: and. They have:
- General hypotenuse.
- (because it is a bisector!)
This means - by angle and hypotenuse. Therefore, the corresponding legs of these triangles are equal! That is.
We proved that the point is equally (or equally) distant from the sides of the angle. Point 1 is dealt with. Now let's move on to point 2.
Why is 2 true?
And let's connect the dots and.
This means that it lies on the bisector!
That's all!
How can all this be applied when solving problems? For example, in problems there is often the following phrase: “A circle touches the sides of an angle...”. Well, you need to find something.
Then you quickly realize that
And you can use equality.
3. Three bisectors in a triangle intersect at one point
From the property of a bisector to be the locus of points equidistant from the sides of an angle, the following statement follows:
How exactly does it come out? But look: two bisectors will definitely intersect, right?
And the third bisector could go like this:
But in reality, everything is much better!
Let's look at the intersection point of two bisectors. Let's call it .
What did we use here both times? Yes paragraph 1, of course! If a point lies on a bisector, then it is equally distant from the sides of the angle.
And so it happened.
But look carefully at these two equalities! After all, it follows from them that and, therefore, .
And now it will come into play point 2: if the distances to the sides of an angle are equal, then the point lies on the bisector...what angle? Look at the picture again:
and are the distances to the sides of the angle, and they are equal, which means the point lies on the bisector of the angle. The third bisector passed through the same point! All three bisectors intersect at one point! And as an additional gift -
Radii inscribed circles.
(To be sure, look at another topic).
Well, now you'll never forget:
The point of intersection of the bisectors of a triangle is the center of the circle inscribed in it.
Let's move on to to the following property... Wow, the bisector has many properties, right? And this is great, because the more properties, the more tools for solving bisector problems.
4. Bisector and parallelism, bisectors of adjacent angles
The fact that the bisector divides the angle in half in some cases leads to completely unexpected results. For example,
Case 1
Great, right? Let's understand why this is so.
On the one hand, we draw a bisector!
But, on the other hand, there are angles that lie crosswise (remember the theme).
And now it turns out that; throw out the middle: ! - isosceles!
Case 2
Imagine a triangle (or look at the picture)
Let's continue the side beyond the point. Now we have two angles:
- - internal corner
- - the outer corner is outside, right?
So, now someone wanted to draw not one, but two bisectors at once: both for and for. What will happen?
Will it work out? rectangular!
Surprisingly, this is exactly the case.
Let's figure it out.
What do you think the amount is?
Of course, - after all, they all together make such an angle that it turns out to be a straight line.
Now remember that and are bisectors and see that inside the angle there is exactly half from the sum of all four angles: and - - that is, exactly. You can also write it as an equation:
So, incredible but true:
The angle between the bisectors of the internal and external angles of a triangle is equal.
Case 3
Do you see that everything is the same here as for the internal and external corners?
Or let's think again why this happens?
Again, as for adjacent corners,
(as corresponding with parallel bases).
And again, they make up exactly half from the sum
Conclusion: If the problem contains bisectors adjacent angles or bisectors relevant angles of a parallelogram or trapezoid, then in this problem certainly a right triangle is involved, or maybe even a whole rectangle.
5. Bisector and opposite side
It turns out that the bisector of an angle of a triangle divides the opposite side not just in some way, but in a special and very interesting way:
That is:
An amazing fact, isn't it?
Now we will prove this fact, but get ready: it will be a little more difficult than before.
Again - exit to “space” - additional formation!
Let's go straight.
For what? We'll see now.
Let's continue the bisector until it intersects with the line.
Is this a familiar picture? Yes, yes, yes, exactly the same as in point 4, case 1 - it turns out that (- bisector)
Lying crosswise
So, that too.
Now let's look at the triangles and.
What can you say about them?
They are similar. Well, yes, their angles are equal as vertical ones. So, in two corners.
Now we have the right to write the relations of the relevant parties.
And now in short notation:
Oh! Reminds me of something, right? Isn't this what we wanted to prove? Yes, yes, exactly that!
You see how great the “spacewalk” proved to be - the construction of an additional straight line - without it nothing would have happened! And so, we have proven that
Now you can safely use it! Let's look at one more property of the bisectors of the angles of a triangle - don't be alarmed, now the hardest part is over - it will be easier.
We get that
Theorem 1:
Theorem 2:
Theorem 3:
Theorem 4:
Theorem 5:
Theorem 6:
Triangle - a polygon with three sides, or a closed broken line with three links, or a figure formed by three segments connecting three points that do not lie on the same straight line (see Fig. 1).
Basic elements of triangle abc
Peaks – points A, B, and C;
Parties – segments a = BC, b = AC and c = AB connecting the vertices;
Angles – α, β, γ formed by three pairs of sides. Angles are often designated in the same way as vertices, with the letters A, B, and C.
The angle formed by the sides of a triangle and lying in its interior area is called an interior angle, and the one adjacent to it is the adjacent angle of the triangle (2, p. 534).
Heights, medians, bisectors and midlines of a triangle
In addition to the main elements in a triangle, other segments with interesting properties are also considered: heights, medians, bisectors and midlines.
Height
Triangle heights- these are perpendiculars dropped from the vertices of the triangle to opposite sides.
To plot the height, you must perform the following steps:
1) draw a straight line containing one of the sides of the triangle (if the height is drawn from the vertex of an acute angle in an obtuse triangle);
2) from the vertex lying opposite the drawn line, draw a segment from the point to this line, making an angle of 90 degrees with it.
The point of intersection of the altitude with the side of the triangle is called height base (see Fig. 2).
Properties of triangle altitudes
In a right triangle, the altitude drawn from the vertex of the right angle splits it into two triangles similar to the original triangle.
In an acute triangle, its two altitudes cut off similar triangles from it.
If the triangle is acute, then all the bases of the altitudes belong to the sides of the triangle, and in an obtuse triangle, two altitudes fall on the continuation of the sides.
Three altitudes in an acute triangle intersect at one point and this point is called orthocenter triangle.
Median
Medians(from Latin mediana – “middle”) - these are segments connecting the vertices of the triangle with the midpoints of the opposite sides (see Fig. 3).
To construct the median, you must perform the following steps:
1) find the middle of the side;
2) connect the point that is the middle of the side of the triangle with the opposite vertex with a segment.
Properties of triangle medians
The median divides a triangle into two triangles of equal area.
The medians of a triangle intersect at one point, which divides each of them in a ratio of 2:1, counting from the vertex. This point is called center of gravity triangle.
The entire triangle is divided by its medians into six equal triangles.
Bisector
Bisectors(from Latin bis - twice and seko - cut) are the straight line segments enclosed inside a triangle that bisect its angles (see Fig. 4).
To construct a bisector, you must perform the following steps:
1) construct a ray coming out from the vertex of the angle and dividing it into two equal parts (the bisector of the angle);
2) find the point of intersection of the bisector of the angle of the triangle with the opposite side;
3) select a segment connecting the vertex of the triangle with the intersection point on the opposite side.
Properties of triangle bisectors
The bisector of an angle of a triangle divides the opposite side in a ratio equal to the ratio of the two adjacent sides.
The bisectors of the interior angles of a triangle intersect at one point. This point is called the center of the inscribed circle.
The bisectors of the internal and external angles are perpendicular.
If the bisector of an exterior angle of a triangle intersects the extension of the opposite side, then ADBD=ACBC.
The bisectors of one internal and two external angles of a triangle intersect at one point. This point is the center of one of the three excircles of this triangle.
The bases of the bisectors of two internal and one external angles of a triangle lie on the same straight line if the bisector of the external angle is not parallel to the opposite side of the triangle.
If the bisectors of the external angles of a triangle are not parallel to opposite sides, then their bases lie on the same straight line.
