Height calculation. Find the greatest height of the triangle

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 that forms 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 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).

How to find the greatest or smallest height of a triangle? The smaller the height of the triangle, the greater the height drawn to it. That is, the greatest of the altitudes of a triangle is the one drawn to its shortest side. - the one drawn to the largest side of the triangle.

To find the greatest height of a triangle , we can divide the area of ​​the triangle by the length of the side to which this height is drawn (that is, by the length of the smallest side of the triangle).

Accordingly, d To find the smallest height of a triangle You can divide the area of ​​a triangle by the length of its longest side.

Task 1.

Find the smallest height of a triangle whose sides are 7 cm, 8 cm and 9 cm.

Given:

AC=7 cm, AB=8 cm, BC=9 cm.

Find: the smallest height of the triangle.

Solution:

The smallest altitude of a triangle is the one drawn to its longest side. This means that we need to find the height AF drawn to side BC.

For convenience of notation, we introduce the notation

BC=a, AC=b, AB=c, AF=ha.

The height of a triangle is equal to the quotient of twice the area of ​​the triangle divided by the side to which this height is drawn. can be found using Heron's formula. That's why

We calculate:

Answer:

Task 2.

Find the longest side of a triangle with sides 1 cm, 25 cm and 30 cm.

Given:

AC=25 cm, AB=11 cm, BC=30 cm.

Find:

greatest altitude of triangle ABC.

Solution:

The greatest height of a triangle is drawn to its shortest side.

This means that we need to find the height CD drawn to side AB.

For convenience, let us denote

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 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.

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 simplest 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 the value of one of the sides of a triangle and the size of the angle between this side and the base are known. Then the formula for calculating the height will look like this: h = b ∙ sin γ + c ∙ sin β.
3. When the area of ​​the triangle is given - 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. In simple terms, 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 the height of a 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. Height value 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 easiest way to measure 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.

Triangles.

Basic concepts.

Triangle is a figure consisting of three segments and three points that do not lie on the same straight line.

The segments are called parties, and the points are peaks.

Sum of angles triangle is 180º.

Height of the triangle.

Triangle height- this is a perpendicular drawn from the vertex to the opposite side.

In an acute triangle, the height is contained within the triangle (Fig. 1).

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

In an obtuse triangle, the altitude extends outside the triangle (Fig. 3).

Properties of the altitude of a triangle:

Bisector of a triangle.

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

Properties of a bisector:

Median of a triangle.

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

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 middle line of the triangle is parallel to the third side and equal to half of it

External angle of a triangle.

External corner of a triangle is equal to the sum of two non-adjacent internal angles (Fig. 11).

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

Right triangle.

Right triangle 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 altitude 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 angles A and B.

Fig.14a

Isosceles triangle.

Isosceles triangle is a triangle whose two sides are equal (Fig. 13).

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

In an isosceles triangle, the base angles 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 altitude 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 unique properties that will help you successfully solve problems involving these shapes. Some of these properties are outlined above. But we repeat them again, adding to them a few other wonderful features:

Signs of equality of triangles.

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.

Second sign of equality: if a side and its adjacent angles of one triangle are equal to the side and its adjacent angles of another triangle, then such triangles are congruent.

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 altitude 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 · A.C. · 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).

A triangle inscribed in a circle.

A triangle is said to be inscribed in a circle if it touches it with all its vertices (Fig. 17 a).

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

Sinus acute angle x opposite leg to hypotenuse.
It is denoted as follows: sinx.

Cosine acute angle x of a right triangle is the ratio adjacent leg to hypotenuse.
Denoted as follows: cos x.

Tangent acute angle x- this is the ratio of the opposite side to the adjacent side.
It is designated as follows: tgx.

Cotangent acute angle x- this is the ratio of the adjacent side to the opposite side.
Designated as follows: ctgx.

Rules:

Leg opposite the 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 by tg x:

b = a tg x

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

a = b· ctg x.

For any acute angle x:

sin (90° - x) = cos x

cos (90° - x) = sin x