How is the height found in a right triangle. Right triangle

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Right triangle - this is a triangle in which one of the angles is straight, that is, equal to 90 degrees.

The side opposite the right angle is called the hypotenuse (in the figure indicated as c or AB)
The side adjacent to the right angle is called the leg. Each right triangle has two legs (in the figure they are designated as a and b or AC and BC)

Formulas and properties of a right triangle

Formula designations:

(see picture above)

a, b- legs of a right triangle

c- hypotenuse

α, β - acute angles of a triangle

S- square

h- height dropped from the top right angle to the hypotenuse

m a a from the opposite corner ( α )

m b- median drawn to the side b from the opposite corner ( β )

m c- median drawn to the side c from the opposite corner ( γ )

IN right triangle any of the legs is less than the hypotenuse(Formula 1 and 2). This property is a consequence of the Pythagorean theorem.

Cosine of any of the acute angles less than one (Formula 3 and 4). This property follows from the previous one. Since any of the legs is less than the hypotenuse, the ratio of leg to hypotenuse is always less than one.

The square of the hypotenuse is equal to the sum of the squares of the legs (Pythagorean theorem). (Formula 5). This property is constantly used when solving problems.

Area of a right triangle equal to half the product of legs (Formula 6)

Sum of squared medians to the legs is equal to five squares of the median to the hypotenuse and five squares of the hypotenuse divided by four (Formula 7). In addition to the above, there is 5 more formulas, therefore, it is recommended that you also read the lesson “Median of a Right Triangle,” which describes the properties of the median in more detail.

Height of a right triangle is equal to the product of the legs divided by the hypotenuse (Formula 8)

The squares of the legs are inversely proportional to the square of the height lowered to the hypotenuse (Formula 9). This identity is also one of the consequences of the Pythagorean theorem.

Hypotenuse length equal to the diameter (two radii) of the circumscribed circle (Formula 10). Hypotenuse of a right triangle is the diameter of the circumcircle. This property is often used in problem solving.

Inscribed radius V right triangle circle can be found as half of the expression including the sum of the legs of this triangle minus the length of the hypotenuse. Or as the product of legs divided by the sum of all sides (perimeter) of a given triangle. (Formula 11)
Sine of angle relation to the opposite this angle leg to hypotenuse(by definition of sine). (Formula 12). This property is used when solving problems. Knowing the sizes of the sides, you can find the angle they form.

The cosine of angle A (α, alpha) in a right triangle will be equal to attitude adjacent this angle leg to hypotenuse(by definition of sine). (Formula 13)

Triangle - This is one of the most famous geometric figures. It is used everywhere - not only in drawings, but also as interior items, parts various designs and buildings. There are several types of this figure - rectangular is one of them. His distinctive feature is the presence of a right angle equal to 90°. To find two of the three heights, it is enough to measure the legs. The third is the value between the vertex of the right angle and the middle of the hypotenuse. Often in geometry the question is how to find the height of a right triangle. Let's solve this simple problem.

Necessary:

- ruler;
– a book on geometry;
- right triangle.

Instructions:

Draw a triangle with a right angle ABC, where is the angle ABC equals 90 ° , that is, it is direct. Lower the height H from a right angle to the hypotenuse - a segment AS. Mark the place where the segments touch with a dot. D.
You should now have another triangle - A.D.B.. Please note that it is similar to the existing one ABC, since the angles ABS And ADB = 90°, then they are equal to each other, and the angle BAD is common to both geometric figures. Correlating them, we can conclude that the parties AD/AB = BD/BS = AB/AS. From the resulting relations it can be concluded that AD equals AB²/AS.
Since the resulting triangle A.D.B. has a right angle, when measuring its sides and hypotenuse, you can use the Pythagorean theorem. Here's what it looks like: AB² = AD² + BD². To solve it, use the resulting equality AD. You should get the following: BD² = AB² - (AB²/AC)². Since the triangle being measured ABS is rectangular, then BS² equals AS² — AB². Therefore, the side BD² equals AB²BC²/AC², which with extraction of the root will be equal to BD = AB*BS/AS.
Similarly, the solution can be derived using another resulting triangle -
BDS. IN in this case, it is also similar to the original ABC, thanks to two angles - ABS And BDS = 90°, and the angle DSB is common. Further, as in the previous example, the proportion is displayed in the aspect ratio, where BD/AB = DS/BS = BS/AS. Hence the value D.S. is derived through equality BS²/AS. Because, AB² = AD*AS , That BS² = DS*AS. From this we conclude that BD² = (AB*BS/AS)² or AD*AS*DS*AS/AS², which equals AD*DS. To find the height in this case it is enough to remove the root from the product D.S. And AD.

It doesn’t matter which school curriculum contains such a subject as geometry. Each of us, as a student, studied this discipline and solved certain problems. But for many people, their school years are behind them and some of the acquired knowledge has been erased from memory.

But what if you suddenly need to find the answer to a certain question from a school textbook, for example, how to find the height in a right triangle? In this case, a modern advanced computer user will first open the Internet and find the information that interests him.

Basic information about triangles

This geometric figure consists of 3 segments connected to each other at the end points, and the points of contact of these points are not on the same straight line. The segments that make up a triangle are called its sides. The junctions of the sides form the tops of the figure, as well as its corners.

Types of triangles depending on angles

This figure can have 3 types of angles: sharp, obtuse and straight. Depending on this, among the triangles the following varieties are distinguished:

Types of triangles depending on the length of the sides

As mentioned earlier, this figure appears from 3 segments. Based on their size, the following types of triangles are distinguished:

How to find the height of a right triangle

Two similar sides of a right triangle that form a right angle at the point of contact are called legs. The segment that connects them is called “hypotenuse”. To find the height in a given geometric figure, you need to lower a line from the top of the right angle to the hypotenuse. With all this, this line should divide the angle at 90? exactly in half. Such a segment is called a bisector.

The picture above shows a right triangle, the height of which we will have to calculate. This can be done in several ways:

If you draw a circle around a triangle and draw a radius, its value will be half the size of the hypotenuse. Based on this, the height of a right triangle can be calculated using the formula:

First of all, a triangle is a geometric figure that is formed by three points that do not lie on the same straight line and are connected by three segments. To find the height of a triangle, you must first determine its type. Triangles differ in the size of their angles and the number of equal angles. According to the size of the angles, a triangle can be acute, obtuse and rectangular. Based on the number of equal sides, triangles are distinguished as isosceles, equilateral and scalene. The altitude is the perpendicular that is lowered to the opposite side of the triangle from its vertex. How to find the height of a triangle?

How to find the height of an isosceles triangle

An isosceles triangle is characterized by equality of sides and angles at its base, therefore the heights of an isosceles triangle drawn to the lateral sides are always equal to each other. Also, the height of this triangle is both a median and a bisector. Accordingly, the height divides the base in half. We consider the resulting right triangle and find the side, that is, the height of the isosceles triangle, using the Pythagorean theorem. Using the following formula, we calculate the height: H = 1/2*√4*a 2 − b 2, where: a is the side side of this isosceles triangle, b is the base of this isosceles triangle.

How to find the height of an equilateral triangle

A triangle with equal sides is called equilateral. The height of such a triangle is derived from the formula for the height of an isosceles triangle. It turns out: H = √3/2*a, where a is the side of this equilateral triangle.

How to find the height of a scalene triangle

A scalene is a triangle in which any two sides are not equal to each other. In such a triangle, all three heights will be different. You can calculate the lengths of the heights using the formula: H = sin60*a = a*(sgrt3)/2, where a is the side of the triangle or first calculate the area of a particular triangle using Heron’s formula, which looks like: S = (p*(p-c)* (p-b)*(p-a))^1/2, where a, b, c are the sides of a scalene triangle, and p is its semi-perimeter. Each height = 2*area/side

How to find the height of a right triangle

A right triangle has one right angle. The height that goes to one of the legs is at the same time the second leg. Therefore, to find the heights lying on the legs, you need to use the modified Pythagorean formula: a = √(c 2 − b 2), where a, b are the legs (a is the leg that needs to be found), c is the length of the hypotenuse. In order to find the second height, you need to put the resulting value a in place of b. To find the third height lying inside the triangle, the following formula is used: h = 2s/a, where h is the height of the right triangle, s is its area, a is the length of the side to which the height will be perpendicular.

A triangle is called acute if all its angles are acute. In this case, all three heights are located inside an acute triangle. A triangle is called obtuse if it has one obtuse angle. Two altitudes of an obtuse triangle are outside the triangle and fall on the continuation of the sides. The third side is inside the triangle. The height is determined using the same Pythagorean theorem.

General formulas for calculating the height of a triangle

Formula for finding the height of a triangle through the sides: H= 2/a √p*(p-c)*(p-b)*(p-b), where h is the height to be found, a, b and c are the sides of a given triangle, p is its semi-perimeter, .
Formula for finding the height of a triangle using an angle and a side: H=b sin y = c sin ß
The formula for finding the height of a triangle through area and side: h = 2S/a, where a is the side of the triangle, and h is the height constructed to side a.
The formula for finding the height of a triangle using the radius and sides: H= bc/2R.

Any school program includes such a subject as geometry. Each of us, as a student, studied this discipline and solved certain problems. But for many people, their school years are behind them and some of the acquired knowledge has been erased from memory.

But what if you suddenly need to find the answer to some question from a school textbook, for example, how to find the height in a right triangle? In this case, the modern advanced computer user will first open the Internet and find the information that interests him.

Basic information about triangles

Types of triangles depending on angles

This figure can have three types of angles: acute, obtuse and straight. Depending on this, the following types of triangles are distinguished:

Types of triangles depending on the length of the sides

As mentioned earlier, this figure is formed from three segments. Based on their size, the following types of triangles are distinguished:

How to find the height of a right triangle

Two identical sides of a right triangle that form a right angle at the point of contact are called legs. The segment that connects them is called the “hypotenuse”. To find the height in a given geometric figure, you need to lower a line from the vertex of the right angle to the hypotenuse. In this case, this line should divide the 90º angle exactly in half. Such a segment is called a bisector.

The picture above shows right triangle, height which we will have to calculate. This can be done in several ways:

If you draw a circle around a triangle and draw a radius, its value will be half the size of the hypotenuse. Based on this, the height of a right triangle can be calculated using the formula:

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