Height of the triangle. Visual Guide (2020). Find the greatest height of a triangle Find the height of a triangle lowered from the vertex

Calculating the height of a triangle depends on the figure itself (isosceles, equilateral, scalene, rectangular). In practical geometry, complex formulas, as a rule, are not found. It is enough to know the general principle of calculations so that it can be universally applicable to all triangles. Today we will introduce you to the basic principles of calculating the height of a figure, calculation formulas based on the properties of the heights of triangles.

What is height?

Height has several distinctive properties

1. The point where all the heights connect is called the orthocenter. If the triangle is pointed, then the orthocenter is located inside the figure; if one of the angles is obtuse, then the orthocenter, as a rule, is located outside.
2. In a triangle where one angle is 90°, the orthocenter and the vertex coincide.
3. Depending on the type of triangle, there are several formulas for finding the height of the triangle.

Traditional Computing

1. If p is half the perimeter, then a, b, c are the designation of the sides of the required figure, h is the height, then the first and most simple formula will look like this: h = 2/a √p(p-a) (p-b) (p-c).
2. IN school textbooks You can often find problems in which you know the value of one of the sides of the triangle and the size of the angle between this side and the base. Then the formula for calculating the height will look like this: h = b ∙ sin γ + c ∙ sin β.
3. When given area of ​​a triangle– S, as well as the length of the base – a, then the calculations will be as simple as possible. The height is found using the formula: h = 2S/a.
4. When the radius of the circle described around the figure is given, we first calculate the lengths of its two sides, and then proceed to calculate the given height of the triangle. To do this, we use the formula: h = b ∙ c/2R, where b and c are the two sides of the triangle that are not the base, and R is the radius.

All sides of this figure are equivalent, their lengths are equal, therefore the angles at the base will also be equal. It follows from this that the heights that we draw on the bases will also be equal, they are also medians and bisectors at the same time. Speaking in simple language, the altitude in an isosceles triangle divides the base in two. The triangle with a right angle, which is obtained after drawing the height, will be considered using the Pythagorean theorem. Let us denote the side as a and the base as b, then the height h = ½ √4 a2 − b2.

How to find the height of an equilateral triangle?

The formula for an equilateral triangle (a figure where all sides are equal in size) can be found based on previous calculations. It is only necessary to measure the length of one of the sides of the triangle and designate it as a. Then the height is derived by the formula: h = √3/2 a.

How to find height right triangle?

As you know, the angle in a right triangle is 90°. The height lowered by one side is also the second side. The altitudes of a triangle with a right angle will lie on them. To obtain data on height, you need to slightly transform the existing Pythagorean formula, designating the legs - a and b, and also measuring the length of the hypotenuse - c.

Let's find the length of the leg (the side to which the height will be perpendicular): a = √ (c2 − b2). The length of the second leg is found using exactly the same formula: b =√ (c2 − b2). After which you can begin to calculate the height of a triangle with a right angle, having first calculated the area of ​​the figure - s. The height value is h = 2s/a.

Calculations with scalene triangle

When a scalene triangle has acute angles, the height lowered to the base is visible. If the triangle has an obtuse angle, then the height may be outside the figure, and you need to mentally continue it to get the connecting point of the height and the base of the triangle. The most in a simple way to measure the height is to calculate it through one of the sides and the size of the angles. The formula is as follows: h = b sin y + c sin ß.

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 It 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 notations
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).

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

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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

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 circumcircle 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)) ),)

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.

First of all, a triangle is a geometric figure that 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 the angles and the number 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 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, the following formula is used: h = 2s/a, where h is the height of the 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.