Find the greatest height of the triangle. Triangle height

First of all, a triangle is geometric figure, which is formed by three points that do not lie on the same straight line and are connected by three segments. To find the height of a triangle, you must first determine its type. Triangles differ in the size of their angles and the number of equal angles. According to the size of the angles, a triangle can be acute, obtuse and rectangular. Based on the number of equal sides, triangles are distinguished as isosceles, equilateral and scalene. The altitude is the perpendicular that is lowered to the opposite side of the triangle from its vertex. How to find the height of a triangle?

How to find the height of an isosceles triangle

An isosceles triangle is characterized by equality of sides and angles at its base, therefore the heights of an isosceles triangle drawn to the lateral sides are always equal to each other. Also, the height of this triangle is both a median and a bisector. Accordingly, the height divides the base in half. We consider the resulting right triangle and find the side, that is, the height of the isosceles triangle, using the Pythagorean theorem. Using the following formula, we calculate the height: H = 1/2*√4*a 2 − b 2, where: a is the side side of this isosceles triangle, b is the base of this isosceles triangle.

How to find the height of an equilateral triangle

A triangle with equal sides is called equilateral. The height of such a triangle is derived from the formula for the height of an isosceles triangle. It turns out: H = √3/2*a, where a is the side of this equilateral triangle.

How to find the height of a scalene triangle

A scalene is a triangle in which any two sides are not equal to each other. In such a triangle, all three heights will be different. You can calculate the lengths of the heights using the formula: H = sin60*a = a*(sgrt3)/2, where a is the side of the triangle or first calculate the area of a particular triangle using Heron’s formula, which looks like: S = (p*(p-c)* (p-b)*(p-a))^1/2, where a, b, c are the sides of a scalene triangle, and p is its semi-perimeter. Each height = 2*area/side

How to find the height of a right triangle

A right triangle has one right angle. The height that goes to one of the legs is at the same time the second leg. Therefore, to find the heights lying on the legs, you need to use the modified Pythagorean formula: a = √(c 2 − b 2), where a, b are the legs (a is the leg that needs to be found), c is the length of the hypotenuse. In order to find the second height, you need to put the resulting value a in place of b. To find the third height lying inside the triangle, use the following formula: h = 2s/a, where h is the height right triangle, s is its area, a is the length of the side to which the height will be perpendicular.

A triangle is called acute if all its angles are acute. In this case, all three heights are located inside an acute triangle. A triangle is called obtuse if it has one obtuse angle. Two altitudes of an obtuse triangle are outside the triangle and fall on the continuation of the sides. The third side is inside the triangle. The height is determined using the same Pythagorean theorem.

General formulas for calculating the height of a triangle

Formula for finding the height of a triangle through the sides: H= 2/a √p*(p-c)*(p-b)*(p-b), where h is the height to be found, a, b and c are the sides of a given triangle, p is its semi-perimeter, .
Formula for finding the height of a triangle using an angle and a side: H=b sin y = c sin ß
The formula for finding the height of a triangle through area and side: h = 2S/a, where a is the side of the triangle, and h is the height constructed to side a.
The formula for finding the height of a triangle using the radius and sides: H= bc/2R.

To solve many geometric problems you need to find the height given figure. These tasks have applied value. 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 create patterns, you need to have an idea of the properties

Many people, despite good grades at school, when constructing ordinary geometric figures, have a question about 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 of construction and calculation.

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

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

Using the given parameters, we construct a triangle.
Let us introduce some 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 angles are a, b, c.
The altitude is the perpendicular drawn 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.
Let's denote the intersection point of the height and side a as H1, and the height itself as 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 will be H3.

Height in a triangle with an obtuse angle

Now let's look at how to find the height of a triangle if there is one (more than 90 degrees). In this case, the altitude drawn from the 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 from the angles α and β, it is necessary to continue the sides of the triangle opposite 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 makes it easier to construct heights and calculate them.

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

Now let’s draw the height from the vertex of angle α, denoting it h1. For this height will be both a bisector and a 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 construct only one height. Thus, to graphically determine how to calculate the height of an isosceles triangle, it is enough to find two of the three heights.

How to find the height of a right triangle

For a right triangle, determining the heights is much easier than for others. This happens because the legs themselves make a right angle, and therefore are heights.

To construct 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 create a triangle in this case, only one construction is required.

When solving various kinds of problems, both of a purely mathematical and applied nature (especially in construction), it is often necessary to determine the value of the height of a certain geometric figure. How to calculate this value (height) in a triangle?

If we combine 3 points in pairs that are not located on a single line, then the resulting figure will be a triangle. Height is the part of a straight line from any vertex of a figure that, when intersecting with the opposite side, forms an angle of 90°.

Find the height of a scalene triangle

Let us determine the value of the height of a triangle in the case when the figure has arbitrary angles and sides.

Heron's formula

h(a)=(2√(p(p-a)*(p-b)*(p-c)))/a, where

p – half the perimeter of the figure, h(a) – a segment to side a, drawn at right angles to it,

p=(a+b+c)/2 – calculation of the semi-perimeter.

If there is an area of the figure, you can use the relation h(a)=2S/a to determine its height.

Trigonometric functions

To determine the length of a segment that makes a right angle when intersecting with side a, you can use the following relations: if side b and angle γ or side c and angle β are known, then h(a)=b*sinγ or h(a)=c *sinβ.
Where:
γ – angle between side b and a,
β is the angle between side c and a.

Relationship with radius

If the original triangle is inscribed in a circle, you can use the radius of such a circle to determine the height. Its center is located at the point where all 3 heights intersect (from each vertex) - the orthocenter, and the distance from it to the vertex (any) is the radius.

Then h(a)=bc/2R, where:
b, c – 2 other sides of the triangle,
R is the radius of the circle circumscribing the triangle.

Find the height in a right triangle

In this type of geometric figure, 2 sides, when intersecting, form a right angle - 90°. Therefore, if you want to determine the height value in it, then you need to calculate either the size of one of the legs, or the size of the segment forming 90° with the hypotenuse. When designating:
a, b – legs,
c – hypotenuse,
h(c) – perpendicular to the hypotenuse.
You can make the necessary calculations using the following relationships:

Pythagorean theorem:

a=√(c 2 -b 2),
b=√(c 2 -a 2),
h(c)=2S/c, because S=ab/2, then h(c)=ab/c.

Trigonometric functions:

a=c*sinβ,
b=c*cosβ,
h(c)=ab/c=с* sinβ* cosβ.

Find the height of an isosceles triangle

This geometric figure is distinguished by the presence of two sides of equal size and a third – the base. To determine the height drawn to the third, distinct side, the Pythagorean theorem comes to the rescue. With notation
a – side,
c – base,
h(c) is a segment to c at an angle of 90°, then h(c)=1/2 √(4a 2 -c 2).

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

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✪ HEIGHT MEDIAN BIsectrix of a triangle Grade 7

✪ bisector, median, altitude of a triangle. Geometry 7th grade

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

✪ Median, bisector, altitude of triangle | Geometry

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

Subtitles

Properties of the point of intersection of three altitudes 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, you 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)))

Point E should be taken as the intersection of two altitudes of the triangle.)

Orthocenter isogonally conjugate to the center circumscribed circle .
Orthocenter lies on the same line as the centroid, the center circumcircle and the center of a circle of nine points (see Euler’s straight line).
Orthocenter of an acute triangle is the center of the 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 the complementary triangle to the first triangle.
The last property can be formulated as follows: The center of the circle circumscribed about the triangle serves orthocenter additional triangle.
Points, symmetrical orthocenter of a triangle with respect to its sides lie on the circumcircle.
Points, symmetrical orthocenter triangles relative 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 circumcircle 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 great as the distance from the center of the circumcircle to the opposite side.
Any segment drawn from orthocenter Before intersecting with the circumcircle, it is always divided in half by the Euler circle. Orthocenter is the homothety center of these two circles.
Hamilton's theorem. Three straight line segments connecting the orthocenter with the vertices of an acute triangle split it into three triangles having the same Euler circle (circle of nine points) as the original acute triangle.
Corollaries of Hamilton's theorem:
- Three straight line segments connecting the orthocenter with the vertices of an acute triangle divide it into three Hamilton triangle having equal radii of circumscribed circles.
- The radii of circumscribed circles of three Hamilton triangles equal to the radius of the circle circumscribed about the original acute triangle.
In an acute triangle, the orthocenter lies inside the triangle; in an obtuse angle - outside the triangle; in a rectangular one - at the vertex of a right angle.

Properties of altitudes of an isosceles triangle

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

Properties of the bases of altitudes of a triangle

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

Other properties of triangle altitudes

If a triangle versatile (scalene), then it internal the 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 triangle there are two heights cut off similar triangles from it.
In a right triangle height, drawn from the vertex of a right angle, splits it into two triangles similar to the original one.

Properties of the minimum altitude of a triangle

The minimum altitude 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 altitudes.
The minimum straight cut in the plane through which a rigid triangular plate can be pulled must have a length equal to the smallest of the heights of this plate.
With the 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 height of the triangle.
The minimum height in a triangle always lies within that triangle.

Basic relationships

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 by 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)- product of the sides, R − (\displaystyle R-) circumscribed circle radius
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)- 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 descends h a (\displaystyle h_(a)).
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.

Right Triangle Altitude Theorem

If the altitude in a right triangle ABC is of length h (\displaystyle h) drawn from the vertex of a right angle, divides the hypotenuse with 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 altitude of a triangle is the perpendicular descended from any vertex of the triangle to the opposite side, or to its extension (the side to which the perpendicular descends is in this case 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 altitudes lie inside the triangle.

In a right triangle, the legs serve as altitudes.

How to find height from base and area

Let us recall the formula for calculating the area of a triangle. The area of a triangle is calculated using 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 - height of the triangle

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

Keep in mind that the base of a triangle can be any side that the height is lowered to (regardless of how the triangle is positioned). To better understand this, imagine that you can rotate this triangle. Turn 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 into the formula to calculate the area (A = 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

Remember the properties of an equilateral triangle. In an equilateral triangle, all sides and all angles are equal (each angle is 60˚). If you draw the height in such a triangle, you will 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 equal to: a2+b2=c2. This theorem can be used to find the height of an equilateral triangle!

Divide the equilateral triangle into two right triangles (to do this, draw the height). Then label 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 the equilateral triangle, and leg “b” is the desired height of the equilateral triangle.

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

Plug 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 meanings you know. You can find the height of a triangle if you know the values of 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 the 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. Using this formula, you will have to perform several steps. First you need to find the variable “s” (we denote half the perimeter of the triangle with this letter). 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, as a second step, 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 to find the area: 1/2bh (or 1/2ah, or 1/2ch).

Now find an equivalent expression for height (h). For our triangle the following equation will be valid: 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 equal to 4, side b is the base.

If, according to the conditions of the problem, two sides and an angle are known, 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 all that remains is 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 table of sines, calculate the value of “h”. In our example, h = 1.928.