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 you need to find the height CD drawn to side AB.
For convenience, let us denote
To solve many geometric problems you need to find the height given figure. These tasks have practical significance. 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
For 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 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, draw a perpendicular connecting the vertex right angle and the opposite side. As a result, in order to create a triangle in this case, only one construction is required.
It is almost never possible to determine all the parameters of a triangle without additional constructions. These constructions are unique graphic characteristics of a triangle, which help determine the size of the sides and angles.
Definition
One of these characteristics is the height of the triangle. Altitude is a perpendicular drawn from the vertex of a triangle to its opposite side. A vertex is one of the three points that, together with the three sides, make up a triangle.
The definition of the height of a triangle may sound like this: the height is the perpendicular drawn from the vertex of the triangle to the straight line containing the opposite side.
This definition sounds more complicated, but it more accurately reflects the situation. The fact is that in an obtuse triangle it is not possible to draw the height inside the triangle. As can be seen in Figure 1, the height in this case is external. In addition, it is not a standard situation to construct the height in a right triangle. In this case, two of the three altitudes of the triangle will pass through the legs, and the third from the vertex to the hypotenuse.
Rice. 1. Height of an obtuse triangle.
Typically, the height of a triangle is designated by the letter h. Height is also indicated in other figures.
How to 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 an isosceles triangle, and then say why the same solution is valid for an equilateral triangle.
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 that must be equal, as well as the height at which certain properties act.
Properties of the altitude of an isosceles triangle drawn to the base:
- The height coincides with the median and bisector
- Divides the base into two equal parts.
We denote the height as ВD. We find DC as half of the base, since the height of point D divides the base in half. DC=4
Height is perpendicular, so BDC is right triangle, and the height of VN is the leg of this triangle.
Let's find the height using the Pythagorean theorem: $$ВD=\sqrt(BC^2-HC^2)=\sqrt(25-16)=3$$
Any equilateral triangle is isosceles, only its base is equal to its 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 area of the triangle and the side to which the height is drawn.
The heights in a triangle are not equal, so for the corresponding side it will be possible to calculate the corresponding height.
Triangle area formula: $$S=(1\over2)*bh$$, where b is side of the triangle, a h is the height drawn to this side. Let's express the height from the formula:
$$h=2*(S\over b)$$
If the area is 15, the side is 5, then the height is $$h=2*(15\over5)=6$$
Through the trigonometric function
The third method is suitable if the side and angle at the base are known. To do this you will have to use the trigonometric function.
Rice. 3. Drawing for the problem.
Angle ВСН=300, and side BC=8. We still have the same right triangle BCH. Let's use sine. Sine is the ratio of the opposite side to the hypotenuse, which means: BH/BC=cos BCH.
The angle is known, as is the side. Let's 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 trigonometric functions for 30.45 and 60 degrees - tabular numbers.
What have we learned?
We learned what the height of a triangle is, what heights there are and how they are designated. Figured it out typical tasks and wrote down three formulas for the height of a triangle.
Test on the topic
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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.
Produce necessary calculations can be done 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).
