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
The height is a perpendicular, which means BDC is a right triangle, and the height BH is a 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.
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 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 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 there are and how they are designated. We figured out typical problems and wrote down three formulas for the height of a triangle.
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how to find the height of a triangle if all three sides are given and got the best answer
Answer from Vusat Jafarov[active]
In short, do this: find the area using the formula S = under the root p*(p-a)*(p-b)*(p-c), p is a half-pyrimeter, we find it like this: 15+13+14= 42, this is a pyrimeter and a half-pyrimeter is half a pyrimeter=21 , And a, b, c are the sides, a=15, b=13, c=14, and we get S= under the root 21*(21-15)*(21-13)*(21-14), we get S= under the root 21*6*8*7, S= root of 7056, S=84!!! now we find the height from the formula S=1/2 base times height, base-CE; 84=1/2*14*h, 84=7*h, h=84/7, h=12. Answer: height=12!!!
Answer from User deleted[newbie]
That's why I sometimes feel low! I'm 19 years old, and I can't solve such a problem for 3rd grade, fucked up! Ashamed!
Answer from Al0253[guru]
Cut, weigh. Divide by the specific gravity of the paper. Divide by the thickness of the paper. Divide by the length of the base of the triangle. The resulting height...
Answer from Engineer[guru]
First, according to Heron, we determine the area of the triangle through its sides.
Well, then you can guess for yourself.
Answer 84
Answer from LILU[active]
The height divides the base into two equal parts, and then use the Pythagorean theorem. But basically, you're lazy.
Answer from IomoN[guru]
Thank you - “I remembered my GOLDEN childhood”))
Answer: the height is 12 cm. And the solution... VERY simple)... No formulas at all)... But according to the Pythagorean theorem.
Draw a triangle... along with the height... You now see 2 triangles “inside the original one”.
The base CE is where point M is located.
If we denote the distance CM=X, then the distance MU=(14-X).
Now we find X if we equate the calculation of the height from these two triangles (the square root on both the left and right sides of the equation - I immediately “remove” it). We get:
15*15-X*X=13*13-(14-X) *(14-X).. . If solved correctly, then SM=X=9 cm.
Then the required height is DM*DM=15*15-9*9=225-81=144.
We take the square root...and DM=12 cm.
Answer from 2 answers[guru]
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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
- 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.
- In a triangle where one angle is 90°, the orthocenter and the vertex coincide.
- Depending on the type of triangle, there are several formulas for finding the height of the triangle.
Traditional Computing
- 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) .
- 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.
- 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.
- 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. 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 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.