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It will almost never be possible to determine all the parameters of a triangle without additional constructions. These constructions are a kind of graphical characteristics of a triangle that help determine the size of the sides and angles.
Definition
One of these characteristics is the height of the triangle. Height is the perpendicular from the top of the triangle to the opposite side. Apex is one of the three points, which, together with the three sides, make up a triangle.
The definition of the height of a triangle can sound like this: the height is a perpendicular drawn from the apex of the triangle to a straight line containing the opposite side.
This definition sounds more complicated, but it reflects the situation more accurately. The fact is that in an obtuse triangle it is impossible to draw the height inside the triangle. As you can see in Figure 1, the height in this case is external. In addition, it is not a standard situation to plot the height in a right-angled triangle. In this case, two of the three heights of the triangle will pass through the legs, and the third from the apex to the hypotenuse.
Rice. 1. Height of an obtuse triangle.
Typically, the height of a triangle is indicated by the letter h. Height is also indicated in other figures.
How do I find the height of a triangle?
There are three standard ways to find the height of a triangle:
Through the Pythagorean theorem
This method is used for equilateral and isosceles triangles. Let's analyze the solution for isosceles triangle, and then let's say why the same decision is valid for an equilateral one.
Given: isosceles triangle ABC with base AC. AB = 5, AC = 8. Find the height of the triangle.
Rice. 2. Drawing for the problem.
For an isosceles triangle, it is important to know which side is the base. This determines the sides, which should be equal, as well as the height at which some properties act.
Properties of the height of an isosceles triangle drawn to the base:
- Height coincides with median and bisector
- Divides the base into two equal parts.
We will designate the height as ВD. We find DС as half from the base, since the height of the point D divides the base in half. DC = 4
The height is a perpendicular, which means that BDC is a right-angled triangle, and the height of BH is the leg of this triangle.
Find the height by the Pythagorean theorem: $$ BD = \ sqrt (BC ^ 2-HC ^ 2) = \ sqrt (25-16) = 3 $$
Any equilateral triangle is isosceles, only its base is equal to the lateral sides. That is, you can use the same procedure.
Through the area of a triangle
This method can be used for any triangle. To use it, you need to know the value of the area of the triangle and the side to which the height is drawn.
The heights in the triangle are not equal, so for the corresponding side it will be possible to calculate the corresponding height.
The formula for the area of a triangle is: $$ S = (1 \ over2) * bh $$, where b is the side of the triangle and h is the height drawn to that side. Let us express the height from the formula:
$$ h = 2 * (S \ over b) $$
If the area is 15, side is 5, then the height $$ h = 2 * (15 \ over5) = 6 $$
Through trigonometric function
The third method is suitable if you know the side and angle at the base. To do this, you will have to use the trigonometric function.
Rice. 3. Drawing for the problem.
Angle BCN = 300, and side BC = 8. We still have the same right-angled triangle BCH. Let's use a sine. Sinus is the ratio of the opposite leg to the hypotenuse, which means: BH / BC = cos BCH.
The corner is known, as is the side. Let us express the height of the triangle:
$$ BH = BC * \ cos (60 \ unicode (xb0)) = 8 * (1 \ over2) = 4 $$
The cosine value is generally taken from the Bradis tables, but the values of the trigonometric functions for 30.45 and 60 degrees are tabular numbers.
What have we learned?
We learned what the height of a triangle is, what heights are and how they are designated. We figured out the typical problems and wrote down three formulas for the height of the triangle.
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When solving various kinds of problems, both of a purely mathematical and applied nature (especially in construction), it is often required to determine the value of the height of a certain geometric figure. How to calculate a given value (height) in a triangle?
If we combine in pairs 3 points that are not on a single straight 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 intersected with the opposite side, forms an angle of 90 °.
Find the height in a versatile triangle
Let us determine the value of the height of the 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 of the perimeter of the figure, h (a) - a segment to side a, drawn at a right angle to it,
p = (a + b + c) / 2 - calculation of the half-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 the segment that makes a right angle when it intersects 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 is the angle between side b and a,
β is the angle between side c and a.
Radius relationship
If the original triangle is inscribed in a circle, you can use the radius of that circle to determine the height. Its center is located at the point where all 3 heights (from each vertex) intersect - 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 geometrical figure, the 2 sides at the intersection form a right angle - 90 °. Therefore, if it is required to determine the value of the height in it, then it is necessary to calculate either the size of one of the legs, or the value of the segment forming 90 ° with the hypotenuse. With the designation:
a, b - legs,
c - hypotenuse,
h (c) - perpendicular to the hypotenuse.
You can make the necessary calculations using the following ratios:
- 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 = c * sinβ * cosβ.
Find the height in an isosceles triangle
This geometric figure is distinguished by the presence of two sides of equal size and the third - the base. To determine the height drawn to the third, excellent side, the Pythagorean theorem comes to the rescue. With the notation
a - lateral 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 do you find the highest or lowest height of a triangle? The lower the height of the triangle, the greater the height drawn to it. That is, the largest of the heights of the triangle is the one that is drawn to its smallest side. - the one that is drawn to the largest of the sides of the triangle.
To find the maximum height of a triangle , the area of a triangle can be divided by the length of the side to which this height is drawn (that is, by the length of the smallest of the sides of the triangle).
Accordingly, d To find the smallest height of a triangle the area of a triangle can be divided by the length of its longest side.
Objective 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 of the heights of the triangle is the one that is drawn to its largest side. So, you need to find the height AF, drawn to the side BC.
For convenience, we introduce the notation
BC = a, AC = b, AB = c, AF = ha.
The height of the triangle is equal to the quotient of dividing the doubled area of the triangle by the side to which this height is drawn. can be found using Heron's formula. So
We calculate:
Answer:
Objective 2.
Find the largest side of a triangle with sides of 1 cm, 25 cm, and 30 cm.
Given:
AC = 25 cm, AB = 11 cm, BC = 30 cm.
Find:
the greatest height of the triangle ABC.
Solution:
The greatest height of a triangle is drawn to its smallest side.
So, you need to find the height CD, drawn to the side AB.
For convenience, we denote
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 the slopes and openings are made. Often, to build patterns, you need to have an idea of \ u200b \ u200bthe properties.
Many people, despite good grades in school, when building normal geometric shapes the question arises of how to find the height of a triangle or parallelogram. Moreover, it is the most difficult. This is because a triangle can be sharp, obtuse, isosceles, or right-angled. Each of them has its own rules of construction and calculation.
How to find the height of a triangle in which all corners are sharp, graphically
If all the angles of the triangle are sharp (each angle in the triangle is less than 90 degrees), then to find the height, you need to do the following.
- According to the specified parameters, we carry out the construction of a triangle.
- Let us introduce the notation. A, B and C will be the tops of the figure. The angles corresponding to each vertex are α, β, γ. The sides opposite to these corners are a, b, c.
- The height is called the perpendicular, lowered from the apex of the angle to the opposite side of the triangle. To find the heights of the triangle, we construct perpendiculars: from the vertex of the angle α to side a, from the vertex of the angle β to side b, and so on.
- The point of intersection of the height and side a is denoted by H1, and the height itself is h1. The intersection point of the height and side b will be H2, the height, respectively, is h2. For side c, the height will be h3, and the intersection point will be H3.
Height in an obtuse triangle
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 height drawn from the obtuse angle will be inside the triangle. The other two heights will be outside the triangle.
Let the angles α and β in our triangle be acute, and the angle γ obtuse. Then, to plot the heights outgoing from the angles α and β, it is necessary to extend the opposite sides of the triangle in order to draw the 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 plot and calculate heights.
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 the height from the vertex of the angle α, we denote it by 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 top of an isosceles triangle and the opposite side, 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, to graphically determine how to calculate the height of an isosceles triangle, it is enough to find two out of three heights.
How to find the height 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 for a triangle in this case, only one construction is required.