Find the maximum height of the triangle. Triangle Height

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Triangle) or pass outside the triangle at an obtuse triangle.

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✪ HEIGHT OF THE MEDIAN BISECTRIX of a triangle Grade 7

✪ bisector, median, triangle height. Geometry Grade 7

✪ Grade 7, lesson 17, Medians, bisectors and heights of a triangle

✪ Median, Bisector, Triangle Height | Geometry

✪ How to find the length of the bisector, median and height? | Chat with me #031 | Boris Trushin

Subtitles

Properties of the point of intersection of three heights of a triangle (orthocenter)

E A → ⋅ B C → + E B → ⋅ C A → + E C → ⋅ A B → = 0 (\displaystyle (\overrightarrow (EA))\cdot (\overrightarrow (BC))+(\overrightarrow (EB))\cdot (\ overrightarrow (CA))+(\overrightarrow (EC))\cdot (\overrightarrow (AB))=0)

(To prove the identity, one should use the formulas

A B → = E B → − E A → , B C → = E C → − E B → , C A → = E A → − E C → (\displaystyle (\overrightarrow (AB))=(\overrightarrow (EB))-(\overrightarrow (EA )),\,(\overrightarrow (BC))=(\overrightarrow (EC))-(\overrightarrow (EB)),\,(\overrightarrow (CA))=(\overrightarrow (EA))-(\overrightarrow (EC)))

The point E should be taken as the intersection of the two heights of the triangle.)

Orthocenter isogonal conjugate to the center circumscribed circle .
Orthocenter lies on the same line as the centroid, the center circumscribed circle and the center of the circle nine points (see the Euler line).
Orthocenter an acute triangle is the center of a circle inscribed in its orthotriangle.
The center of a triangle described by the orthocenter with vertices at the midpoints of the sides of the given triangle. The last triangle is called an additional triangle with respect to the first triangle.
The last property can be formulated as follows: The center of a circle circumscribed about a triangle serves orthocenter additional triangle.
Points, symmetrical orthocenter triangle with respect to its sides lie on the circumscribed circle.
Points, symmetrical orthocenter triangles with respect to the midpoints of the sides also lie on the circumscribed circle and coincide with points diametrically opposite to the corresponding vertices.
If O is the center of the circumscribed circle ΔABC, then O H → = O A → + O B → + O C → (\displaystyle (\overrightarrow (OH))=(\overrightarrow (OA))+(\overrightarrow (OB))+(\overrightarrow (OC))) ,
The distance from the vertex of the triangle to the orthocenter is twice as large as the distance from the center of the circumscribed circle to the opposite side.
Any segment drawn from orthocenter always bisects the Euler circle until it intersects the circumcircle. Orthocenter is the center of the homothety of these two circles.
Theorem Hamilton. Three line segments connecting the orthocenter with the vertices of an acute-angled triangle divide it into three triangles having the same Euler circle (circle of nine points) as the original acute-angled triangle.
Corollaries of Hamilton's theorem:
- Three line segments connecting the orthocenter with the vertices of an acute-angled triangle divide it into three Hamilton triangle having equal radii of circumscribed circles.
- The radii of the circumscribed circles of the three Hamilton triangles are equal to the radius of the circle circumscribed about the original acute-angled triangle.
In an acute triangle, the orthocenter lies inside the triangle; in obtuse - outside the triangle; in a rectangular one - at the vertex of a right angle.

Properties of heights of an isosceles triangle

If in a triangle two heights are equal, then the triangle is isosceles (the Steiner-Lemus theorem), and the third height is both the median and the bisector of the angle from which it emerges.
The converse is also true: in an isosceles triangle, two heights are equal, and the third height is both a median and a bisector.
An equilateral triangle has all three altitudes equal.

Properties of the bases of the heights of a triangle

Foundations heights form the so-called orthotriangle, which has its own properties.
The circle circumscribed near the orthotriangle is the Euler circle. Three midpoints of the sides of the triangle and three midpoints of the three segments connecting the orthocenter with the vertices of the triangle also lie on this circle.
Another formulation of the last property:
- Euler's theorem for a circle nine points. Foundations three heights arbitrary triangle, the midpoints of its three sides ( foundations of its internal medians) and the midpoints of the three segments connecting its vertices with the orthocenter, all lie on the same circle (on nine point circle).
Theorem. In any triangle, the line segment connecting grounds two heights triangle cuts off a triangle similar to the given one.
Theorem. In a triangle, the line segment connecting grounds two heights triangles on two sides antiparallel a third party with whom he has no common points. Through its two ends, as well as through two vertices of the third mentioned side, it is always possible to draw a circle.

Other properties of triangle heights

If triangle versatile (scalene), then its internal bisector drawn from any vertex lies between internal median and height drawn from the same vertex.
The height of a triangle is isogonally conjugate to the diameter (radius) circumscribed circle drawn from the same vertex.
In an acute-angled triangle, two heights cut off similar triangles from it.
In a rectangular triangle height, drawn from the vertex of the right angle , divides it into two triangles similar to the original one.

Properties of the minimum height of a triangle

The minimum height of a triangle has many extreme properties. For example:

The minimum orthogonal projection of a triangle onto lines lying in the plane of the triangle has a length equal to the smallest of its heights.
The minimum straight cut in the plane through which an inflexible triangular plate can be pulled must have a length equal to the smallest of the heights of this plate.
With continuous movement of two points along the perimeter of the triangle towards each other, the maximum distance between them during the movement from the first meeting to the second cannot be less than the length of the smallest of the heights of the triangle.
The minimum height in a triangle is always within that triangle.

Basic ratios

h a = b ⋅ sin ⁡ γ = c ⋅ sin ⁡ β , (\displaystyle h_(a)=b(\cdot )\sin \gamma =c(\cdot )\sin \beta ,)
h a = 2 ⋅ S a , (\displaystyle h_(a)=(\frac (2(\cdot )S)(a))) where S (\displaystyle S)- area of a triangle, a (\displaystyle a)- the length of the side of the triangle on which the height is lowered.
h a = b ⋅ c 2 ⋅ R , (\displaystyle h_(a)=(\frac (b(\cdot )c)(2(\cdot )R)),) where b ⋅ c (\displaystyle b(\cdot )c)- the product of the sides, R − (\displaystyle R-) radius of the circumscribed circle
h a: h b: h c = 1 a: 1 b: 1 c = (b ⋅ c) : (a ⋅ c) : (a ⋅ b) . (\displaystyle h_(a):h_(b):h_(c)=(\frac (1)(a)):(\frac (1)(b)):(\frac (1)(c)) =(b(\cdot )c):(a(\cdot )c):(a(\cdot )b).)
1 h a + 1 h b + 1 h c = 1 r (\displaystyle (\frac (1)(h_(a)))+(\frac (1)(h_(b)))+(\frac (1)(h_ (c)))=(\frac (1)(r))), where r (\displaystyle r) is the radius of the inscribed circle.
S = 1 (1 h a + 1 h b + 1 h c) ⋅ (1 h a + 1 h b − 1 h c) ⋅ (1 h a + 1 h c − 1 h b) ⋅ (1 h b + 1 h c − 1 h a) (\displaystyle S =(\frac (1)(\sqrt (((\frac (1)(h_(a)))+(\frac (1)(h_(b)))+(\frac (1)(h_(c ))))(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(b)))-(\frac (1)(h_(c))) )(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(c)))-(\frac (1)(h_(b))))(\ cdot )((\frac (1)(h_(b)))+(\frac (1)(h_(c)))-(\frac (1)(h_(a)))))))), where S (\displaystyle S)- area of a triangle.
a = 2 h a ⋅ (1 h a + 1 h b + 1 h c) ⋅ (1 h a + 1 h b − 1 h c) ⋅ (1 h a + 1 h c − 1 h b) ⋅ (1 h b + 1 h c − 1 h a) (\ displaystyle a=(\frac (2)(h_(a)(\cdot )(\sqrt (((\frac (1)(h_(a)))+(\frac (1)(h_(b))) +(\frac (1)(h_(c))))(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(b)))-(\ frac (1)(h_(c))))(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(c)))-(\frac (1 )(h_(b))))(\cdot )((\frac (1)(h_(b)))+(\frac (1)(h_(c)))-(\frac (1)(h_ (a))))))))), a (\displaystyle a)- the side of the triangle to which the height falls h a (\displaystyle h_(a)).
The height of an isosceles triangle lowered to the base: h c = 1 2 ⋅ 4 a 2 − c 2 , (\displaystyle h_(c)=(\frac (1)(2))(\cdot )(\sqrt (4a^(2)-c^(2)) ))

where c (\displaystyle c)- base, a (\displaystyle a)- side.

Theorem on the height of a right triangle

If the height in a right triangle ABC is h (\displaystyle h), drawn from the vertex of a right angle, divides the hypotenuse with a length c (\displaystyle c) into segments m (\displaystyle m) and n (\displaystyle n) corresponding to the legs b (\displaystyle b) and a (\displaystyle a), then the following equalities are true.

The height of a triangle is the perpendicular dropped from any vertex of the triangle to the opposite side, or to its extension (the side on which the perpendicular falls, in this case is called the base of the triangle).

In an obtuse triangle, two altitudes fall on the extension of the sides and lie outside the triangle. The third is inside the triangle.

In an acute triangle, all three heights lie inside the triangle.

In a right triangle, the legs serve as heights.

How to find height from base and area

Recall the formula for calculating the area of a triangle. The area of a triangle is calculated by the formula: A=1/2bh.

A is the area of the triangle
b is the side of the triangle on which the height is lowered.
h is the height of the triangle

Look at the triangle and think about what quantities you already know. If you are given an area, label it with the letter "A" or "S". You should also be given the value of the side, designate it with the letter "b". If you are not given an area and you are not given a side, use another method.

Keep in mind that the base of a triangle can be any side of the triangle where the height is dropped (regardless of how the triangle is positioned). To better understand this, imagine that you can rotate this triangle. Rotate it so that the side you know is facing down.

For example, the area of a triangle is 20 and one of its sides is 4. In this case, “‘A = 20″‘, ‘”b = 4′”.

Substitute the values given to you in the formula for calculating the area (A \u003d 1 / 2bh) and find the height. First multiply the side (b) by 1/2, and then divide the area (A) by the resulting value. This way you will find the height of the triangle.

In our example: 20 = 1/2(4)h

20 = 2h
10 = h

Recall the properties of an equilateral triangle. In an equilateral triangle, all sides and all angles are equal (each angle is 60˚). If you draw a height in such a triangle, you get two equal right triangles.
For example, consider an equilateral triangle with side 8.

Remember the Pythagorean theorem. The Pythagorean theorem states that in any right triangle with legs "a" and "b" the hypotenuse "c" is: a2 + b2 \u003d c2. This theorem can be used to find the height of an equilateral triangle!

Divide an equilateral triangle into two right-angled triangles (to do this, draw a height). Then mark the sides of one of the right triangles. The lateral side of an equilateral triangle is the hypotenuse "c" of a right triangle. Leg "a" is equal to 1/2 of the side of an equilateral triangle, and leg "b" is the required height of an equilateral triangle.

So, in our example with an equilateral triangle with a known side equal to 8: c = 8 and a = 4.

Substitute these values into the Pythagorean theorem and calculate b2. First, square "c" and "a" (multiply each value by itself). Then subtract a2 from c2.

42 + b2 = 82
16 + b2 = 64
b2 = 48

Take the square root of b2 to find the height of the triangle. To do this, use a calculator. The resulting value will be the height of your equilateral triangle!

b = √48 = 6.93

How to find height using angles and sides

Think about what values you know. You can find the height of a triangle if you know the sides and angles. For example, if the angle between the base and the side is known. Or if the values of all three sides are known. So, let's denote the sides of the triangle: "a", "b", "c", the angles of the triangle: "A", "B", "C", and the area - the letter "S".

If you know all three sides, you will need the area of the triangle and Heron's formula.

If you know two sides and the angle between them, you can use the following formula to find the area: S=1/2ab(sinC).

If you are given the values of all three sides, use Heron's formula. This formula will require several steps. First you need to find the variable "s" (we will denote by this letter half the perimeter of the triangle). To do this, substitute the known values into this formula: s = (a+b+c)/2.

For a triangle with sides a = 4, b = 3, c = 5, s = (4+3+5)/2. The result is: s=12/2, where s=6.

Then, with the second action, we find the area (the second part of Heron's formula). Area = √(s(s-a)(s-b)(s-c)). Instead of the word "area", insert the equivalent formula for finding the area: 1/2bh (or 1/2ah, or 1/2ch).

Now find the equivalent expression for height (h). The following equation will be valid for our triangle: 1/2(3)h = (6(6-4)(6-3)(6-5)). Where 3/2h=√(6(2(3(1))). It turns out that 3/2h = √(36). Using a calculator, calculate the square root. In our example: 3/2h = 6. It turns out that the height (h) is 4, side b is the base.

If two sides and an angle are known by the condition of the problem, you can use a different formula. Replace the area in the formula with the equivalent expression: 1/2bh. Thus, you will get the following formula: 1/2bh = 1/2ab(sinC). It can be simplified to the following form: h = a(sin C) to remove one unknown variable.

Now it remains to solve the resulting equation. For example, let "a" = 3, "C" = 40 degrees. Then the equation will look like this: "h" = 3(sin 40). Using a calculator and a sine table, calculate the value of "h". In our example, h = 1.928.

To solve many geometric problems, you need to find the height of a given figure. These tasks are of practical importance. When carrying out construction work, determining the height helps to calculate the required amount of materials, as well as determine how accurately slopes and openings are made. Often, to build patterns, you need to have an idea about the properties

Many people, despite good grades at school, when constructing ordinary geometric figures, the question arises of how to find the height of a triangle or parallelogram. And it is the most difficult. This is because a triangle can be acute, obtuse, isosceles, or right. Each of them has its own rules for construction and calculation.

How to find the height of a triangle in which all angles are acute, graphically

If all the angles of the triangle are acute (each angle in the triangle is less than 90 degrees), then to find the height, do the following.

According to the given parameters, we construct a triangle.
Let us introduce notation. A, B and C will be the vertices of the figure. The angles corresponding to each vertex are α, β, γ. The sides opposite these corners are a, b, c.
The height is the perpendicular from the vertex of the angle to the opposite side of the triangle. To find the heights of a triangle, we construct perpendiculars: from the vertex of angle α to side a, from the vertex of angle β to side b, and so on.
The intersection point of the height and side a will be denoted by H1, and the height itself will be h1. The intersection point of the height and side b will be H2, the height, respectively, h2. For side c, the height will be h3 and the intersection point H3.

Height in a triangle with an obtuse angle

Now consider how to find the height of a triangle if one (greater than 90 degrees). In this case, the height drawn from an obtuse angle will be inside the triangle. The remaining two heights will be outside the triangle.

Let the angles α and β in our triangle be acute, and the angle γ be obtuse. Then, to construct the heights coming out of the angles α and β, it is necessary to continue the sides of the triangle opposite to them in order to draw perpendiculars.

How to find the height of an isosceles triangle

Such a figure has two equal sides and a base, while the angles at the base are also equal to each other. This equality of sides and angles facilitates the construction of heights and their calculation.

First, let's draw the triangle itself. Let the sides b and c, as well as the angles β, γ be respectively equal.

Now let's draw a height from the vertex of the angle α, denote it h1. For this height will be both the bisector and the median.

Only one construction can be made for the foundation. For example, draw a median - a segment connecting the vertex of an isosceles triangle and the opposite side, the base, to find the height and bisector. And to calculate the length of the height for the other two sides, you can build only one height. Thus, in order to graphically determine how to calculate the height of an isosceles triangle, it is enough to find two heights out of three.

How to find the height of a right triangle

It is much easier to determine the heights of a right triangle than others. This is because the legs themselves make up a right angle, which means they are heights.

To build the third height, as usual, a perpendicular is drawn connecting the vertex of the right angle and the opposite side. As a result, in order to make a triangle in this case, only one construction is required.

Triangles.

Basic concepts.

Triangle- this is a figure consisting of three segments and three points that do not lie on one straight line.

The segments are called parties, and the points peaks.

Sum of angles triangle is equal to 180 º.

The height of the triangle.

Triangle Height is a perpendicular drawn from a vertex to the opposite side.

In an acute-angled triangle, the height is contained inside the triangle (Fig. 1).

In a right triangle, the legs are the heights of the triangle (Fig. 2).

In an obtuse triangle, the height passes outside the triangle (Fig. 3).

Triangle height properties:

Bisector of a triangle.

Bisector of a triangle- this is a segment that bisects the corner of the vertex and connects the vertex to a point on the opposite side (Fig. 5).

Bisector properties:

The median of a triangle.

Triangle median- this is a segment connecting the vertex with the middle of the opposite side (Fig. 9a).

The length of the median can be calculated using the formula:

2b 2 + 2c 2 - a 2
m a 2 = ——————
4

where m a- median drawn to the side a.

In a right triangle, the median drawn to the hypotenuse is half the hypotenuse:

c
mc = —
2

where mc is the median drawn to the hypotenuse c(Fig. 9c)

The medians of a triangle intersect at one point (at the center of mass of the triangle) and are divided by this point in a ratio of 2:1, counting from the top. That is, the segment from the vertex to the center is twice the segment from the center to the side of the triangle (Fig. 9c).

The three medians of a triangle divide it into six triangles of equal area.

The middle line of the triangle.

Middle line of the triangle- this is a segment connecting the midpoints of its two sides (Fig. 10).

The midline of a triangle is parallel to the third side and equal to half of it.

The outer corner of the triangle.

outside corner triangle is equal to the sum of two non-adjacent interior angles (Fig. 11).

The exterior angle of a triangle is greater than any non-adjacent angle.

Right triangle.

Right triangle- this is a triangle that has a right angle (Fig. 12).

The side of a right triangle opposite the right angle is called hypotenuse.

The other two sides are called legs.

Proportional segments in a right triangle.

1) In a right triangle, the height drawn from the right angle forms three similar triangles: ABC, ACH and HCB (Fig. 14a). Accordingly, the angles formed by the height are equal to the angles A and B.

Fig.14a

Isosceles triangle.

Isosceles triangle- this is a triangle in which two sides are equal (Fig. 13).

These equal sides are called sides, and the third basis triangle.

In an isosceles triangle, the angles at the base are equal. (In our triangle, angle A is equal to angle C).

In an isosceles triangle, the median drawn to the base is both the bisector and the height of the triangle.

Equilateral triangle.

An equilateral triangle is a triangle in which all sides are equal (Fig. 14).

Properties of an equilateral triangle:

Remarkable properties of triangles.

Triangles have original properties that will help you successfully solve problems associated with these shapes. Some of these properties are outlined above. But we repeat them again, adding a few other great features to them:

1) In a right triangle with angles 90º, 30º and 60º, the leg b, lying opposite the angle of 30º, is equal to half of the hypotenuse. A lega more legb√3 times (Fig. 15 a). For example, if the leg of b is 5, then the hypotenuse c necessarily equal to 10, and the leg a equals 5√3.

2) In a right-angled isosceles triangle with angles of 90º, 45º and 45º, the hypotenuse is √2 times the leg (Fig. 15 b). For example, if the legs are 5, then the hypotenuse is 5√2.

3) The middle line of the triangle is equal to half of the parallel side (Fig. 15 with). For example, if the side of a triangle is 10, then the midline parallel to it is 5.

4) In a right triangle, the median drawn to the hypotenuse is equal to half of the hypotenuse (Fig. 9c): mc= c/2.

5) The medians of a triangle, intersecting at one point, are divided by this point in a ratio of 2:1. That is, the segment from the vertex to the point of intersection of the medians is twice the segment from the point of intersection of the medians to the side of the triangle (Fig. 9c)

6) In a right triangle, the midpoint of the hypotenuse is the center of the circumscribed circle (Fig. 15 d).

Signs of equality of triangles.

The first sign of equality: If two sides and the angle between them of one triangle are equal to two sides and the angle between them of another triangle, then such triangles are congruent.

The second sign of equality: if the side and angles adjacent to it of one triangle are equal to the side and angles adjacent to it of another triangle, then such triangles are congruent.

The third sign of equality: If three sides of one triangle are equal to three sides of another triangle, then such triangles are congruent.

Triangle Inequality.

In any triangle, each side is less than the sum of the other two sides.

Pythagorean theorem.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs:

c 2 = a 2 + b 2 .

Area of a triangle.

1) The area of a triangle is equal to half the product of its side and the height drawn to this side:

Ah
S = ——
2

2) The area of a triangle is equal to half the product of any two of its sides and the sine of the angle between them:

1
S = — AB · AC · sin A
2

A triangle circumscribed about a circle.

A circle is called inscribed in a triangle if it touches all its sides (Fig. 16 a).

Triangle inscribed in a circle.

A triangle is called inscribed in a circle if it touches it with all vertices (Fig. 17 a).

Sine, cosine, tangent, cotangent of an acute angle of a right triangle (Fig. 18).

Sinus acute angle x opposite catheter to the hypotenuse.
Denoted like this: sinx.

Cosine acute angle x right triangle is the ratio adjacent catheter to the hypotenuse.
It is denoted as follows: cos x.

Tangent acute angle x is the ratio of the opposite leg to the adjacent leg.
Denoted like this: tgx.

Cotangent acute angle x is the ratio of the adjacent leg to the opposite leg.
Denoted like this: ctgx.

Rules:

Leg opposite corner x, is equal to the product of the hypotenuse and sin x:

b=c sin x

Leg adjacent to the corner x, is equal to the product of the hypotenuse and cos x:

a = c cos x

Leg opposite corner x, is equal to the product of the second leg and tg x:

b = a tg x

Leg adjacent to the corner x, is equal to the product of the second leg and ctg x:

a = b ctg x.

For any acute angle x:

sin (90° - x) = cos x

cos (90° - x) = sin x