When solving a different kind of tasks, both a purely mathematical and applied character (especially in construction), it is often necessary to determine the height value of a certain geometric shape. How to calculate this amount (height) in a triangle?
If we are in pairwise compatible 3 points, located not on a single straight line, then the resulting figure will be a triangle. The height is part of the straight line of any vertex of the figure, which when crossing with the opposite side, forms an angle of 90 °.
Find a height in a versatile triangle
We define the value of the height of the triangle in the case when the figure has arbitrary corners and parties.
Formula Gerona
h (a) \u003d (2√ (p (p-a) * (p-b) * (P-C))) / a, where
p is half the perimeter of the figure, H (a) - cut to the side A, spent at right angles to it,
B, C - 2 Other triangle sides,
P \u003d (A + B + C) / 2 - Calculation of half-version.
In the case of the area of \u200b\u200bthe figure to determine its height, it is possible to use the ratio H (a) \u003d 2s / a.
Trigonometric functions
To determine the length of the segment, which is when intersection with a side A, a straight angle can be used by the following ratios: if the side B is known and the angle γ or the side C and the angle β, then h (a) \u003d b * sinγ or h (a) \u003d c * sinβ.
Where:
γ is the angle between the side B and A,
β is the angle between C and a.
Relationship with radius
If the initial triangle is entered into a circle, to determine the size of the height, you can use the radius of such a circle. Its center is located at the point where all 3 heights are intersect (from each vertex) - an orthocentre, and the distance from it to the top (any) is radius.
Then h (a) \u003d BC / 2R, where:
B, C - 2 Other triangle sides,
R is a radius describing triangle circumference.
Find a height in a rectangular triangle
In this form, the geometric shape of 2 sides with the intersection form a straight angle - 90 °. Therefore, if it is required to determine in it the value of the height, then it is necessary to calculate either the size of one of the cathets, or the amount of the segment forming with a hypotenurium 90 °. When designation:
A, B - Kartets,
C - hypotenuse,
h (c) - perpendicular on the hypotenuse.
It is possible to produce the necessary calculations using the following ratios:
- Pytagorova Theorem:
a \u003d √ (c 2-b. 2 ),
B \u003d √ (C 2-a. 2 ),
H (C) \u003d 2S / C, because S \u003d AB / 2, then H (C) \u003d AB / C.
- Trigonometric functions:
a \u003d c * sinβ
B \u003d C * Cosβ,
H (C) \u003d AB / C \u003d C * SINβ * COSβ.
Find a height in an equally traded triangle
This geometric shape is characterized by the presence of two sides of equal size and third - base. To determine the height spent on the third, excellent side, the Pythagora theorem comes to the aid. With the notation
a - side,
C is the basis
h (c) - segment to C at an angle of 90 °, then h (c) \u003d 1/2 √ (4a 2-c. 2 ).
Find the height of the triangle of the equilateral
In such a triangle, the equality of all sides is noted, and the angles are 60 °. Based on the formula for finding a perpendicular to the base for an equilibrium triangle, we obtain the following ratio, which is valid for all three heights.
h \u003d √3a / 2.
0
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