The video course "Get an A" includes all the topics you need to successful delivery USE in mathematics for 60-65 points. Completely all tasks 1-13 profile exam mathematics. Also suitable for passing the Basic USE in mathematics. If you want to pass the exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!

Preparation course for the exam for grades 10-11, as well as for teachers. Everything you need to solve part 1 of the exam in mathematics (the first 12 problems) and problem 13 (trigonometry). And this is more than 70 points on the Unified State Examination, and neither a hundred-point student nor a humanist can do without them.

All the necessary theory. Quick solutions, traps and secrets of the exam. All relevant tasks of part 1 from the Bank of FIPI tasks have been analyzed. The course fully complies with the requirements of the USE-2018.

The course contains 5 large topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.

Hundreds of exam tasks. Text problems and probability theory. Simple and easy to remember problem solving algorithms. Geometry. Theory, reference material, analysis of all types of USE tasks. Stereometry. Cunning tricks for solving, useful cheat sheets, development of spatial imagination. Trigonometry from scratch - to task 13. Understanding instead of cramming. Visual explanation of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. Base for solving complex problems of the 2nd part of the exam.

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 a given value (height) in a triangle?

If we combine 3 points in pairs that are not located on a single straight line, then the resulting figure will be a triangle. An altitude is the part of a line from any vertex of a figure that, when intersected with the opposite side, forms an angle of 90°.

Find the height in a scalene 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) - segment to side a, drawn at right angles to it,

p=(a+b+c)/2 – calculation of the half-perimeter.

If there is an area of ​​the figure, to determine its height, you can use the ratio h(a)=2S/a.

Trigonometric functions

To determine the length of a segment that makes a right angle at the intersection with side a, you can use the following relationships: 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.

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 describing the triangle.

Find the height in a right triangle

In this form of a geometric figure, 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 that forms 90 ° with the hypotenuse. When designating:
a, b - legs,
c is the hypotenuse,
h(c) is the perpendicular to the hypotenuse.
Produce necessary calculations can be done using the following relations:

  • Pythagorean theorem:

a \u003d √ (c 2 -b 2),
b \u003d √ (c 2 -a 2),
h(c)=2S/c 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 altitude in an isosceles triangle

This geometric figure has two sides equal size and the third is the foundation. To determine the height drawn to the third, different side, the Pythagorean theorem comes to the rescue. With the designations
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).


First of all, a triangle is a geometric figure, which is formed by three points that do not lie on one straight line, which are connected by three segments. To find what the height of a triangle is, it is necessary, first of all, to determine its type. Triangles differ in the size of the angles and the number of equal angles. According to the size of the angles, the triangle can be acute-angled, obtuse-angled and right-angled. According to the number of equal sides, isosceles, equilateral and scalene triangles are distinguished. The height 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 the equality of sides and angles at its base, therefore, the heights of an isosceles triangle drawn to the 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 \u003d 1/2 * √4 * a 2 - b 2, where: a - the side of this isosceles triangle, b - the base of this isosceles triangle.

How to find the height of an equilateral triangle

A triangle with equal sides is called an equilateral triangle. 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 the given equilateral triangle.

How to find the height of a scalene triangle

A scalene triangle is a triangle in which no two sides are equal to each other. In such a triangle, all three heights will be different. You can calculate the height lengths using the formula: H \u003d sin60 * a \u003d a * (sgrt3) / 2, where a is the side of the triangle, or first calculate the area of ​​​​a particular triangle using the Heron formula, which looks like: S \u003d (p * (p-c) * (p-b)*(p-a))^1/2, where a, b, c are sides of a scalene triangle, and p is its half-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 passes 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 \u003d √ (c 2 - b 2), where a, b are the legs (a is the leg 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 \u003d 2s / a, where h is the height of a right-angled 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 extension of the sides. The third side is inside the triangle. The height is determined using the same Pythagorean theorem.

General formulas like calculating the height of a triangle

  • The 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 the given triangle, p is its semi-perimeter, .
  • The formula for finding the height of a triangle in terms of angle and side: H=b sin y = c sin ß
  • The formula for finding the height of a triangle in terms of area and side: h = 2S / a, where a is the side of the triangle, and h is the height built to side a.
  • The formula for finding the height of a triangle in terms of radius and sides: H= bc/2R.

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The height of a triangle is the perpendicular dropped from any vertex of the triangle to the opposite side, or to its extension (the side on which the perpendicular falls, in this case is called the base of the triangle).

In an obtuse triangle, two altitudes fall on the extension of the sides and lie outside the triangle. The third is inside the triangle.

In an acute triangle, all three heights lie inside the triangle.

In a right triangle, the legs serve as heights.

How to find height from base and area

Recall the formula for calculating the area of ​​a triangle. The area of ​​a triangle is calculated by the formula: A=1/2bh.

  • A is the area of ​​the triangle
  • b is the side of the triangle on which the height is lowered.
  • h is the height of the triangle

Look at the triangle and think about what quantities you already know. If you are given an area, label it with the letter "A" or "S". You should also be given the value of the side, designate it with the letter "b". If you are not given an area and you are not given a side, use another method.

Keep in mind that the base of a triangle can be any side of the triangle where the height is dropped (regardless of how the triangle is positioned). To better understand this, imagine that you can rotate this triangle. Rotate it so that the side you know is facing down.

For example, the area of ​​a triangle is 20 and one of its sides is 4. In this case, “‘A = 20″‘, ‘”b = 4′”.

Substitute the values ​​given to you in the formula for calculating the area (A \u003d 1 / 2bh) and find the height. First multiply the side (b) by 1/2, and then divide the area (A) by the resulting value. This way you will find the height of the triangle.

In our example: 20 = 1/2(4)h

20 = 2h
10 = h

Recall the properties of an equilateral triangle. In an equilateral triangle, all sides and all angles are equal (each angle is 60˚). If you draw a height in such a triangle, you get two equal right triangles.
For example, consider an equilateral triangle with side 8.

Remember the Pythagorean theorem. The Pythagorean theorem states that in any right triangle with legs "a" and "b" the hypotenuse "c" is: a2 + b2 \u003d c2. This theorem can be used to find the height of an equilateral triangle!

Divide an equilateral triangle into two right-angled triangles (to do this, draw a height). Then mark the sides of one of the right triangles. The lateral side of an equilateral triangle is the hypotenuse "c" of a right triangle. Leg "a" is equal to 1/2 of the side of an equilateral triangle, and leg "b" is the required height of an equilateral triangle.

So, in our example with an equilateral triangle with a known side equal to 8: c = 8 and a = 4.

Substitute these values ​​into the Pythagorean theorem and calculate b2. First, square "c" and "a" (multiply each value by itself). Then subtract a2 from c2.

42 + b2 = 82
16 + b2 = 64
b2 = 48

Extract Square root from b2 to find the height of the triangle. To do this, use a calculator. The resulting value will be the height of your equilateral triangle!

b = √48 = 6.93

How to find height using angles and sides

Think about what values ​​you know. You can find the height of a triangle if you know the sides and angles. For example, if the angle between the base and the side is known. Or if the values ​​of all three sides are known. So, let's denote the sides of the triangle: "a", "b", "c", the angles of the triangle: "A", "B", "C", and the area - the letter "S".

If you know all three sides, you will need the area of ​​the triangle and Heron's formula.

If you know two sides and the angle between them, you can use the following formula to find the area: S=1/2ab(sinC).

If you are given the values ​​of all three sides, use Heron's formula. This formula will require several steps. First you need to find the variable "s" (we will denote by this letter half the perimeter of the triangle). To do this, substitute the known values ​​into this formula: s = (a+b+c)/2.

For a triangle with sides a = 4, b = 3, c = 5, s = (4+3+5)/2. The result is: s=12/2, where s=6.

Then, with the second action, we find the area (the second part of Heron's formula). Area = √(s(s-a)(s-b)(s-c)). Instead of the word "area", insert the equivalent formula for finding the area: 1/2bh (or 1/2ah, or 1/2ch).

Now find the equivalent expression for height (h). The following equation will be valid for our triangle: 1/2(3)h = (6(6-4)(6-3)(6-5)). Where 3/2h=√(6(2(3(1))). It turns out that 3/2h = √(36). Using a calculator, calculate the square root. In our example: 3/2h = 6. It turns out that the height (h) is 4, side b is the base.

If two sides and an angle are known by the condition of the problem, you can use a different formula. Replace the area in the formula with the equivalent expression: 1/2bh. Thus, you will get the following formula: 1/2bh = 1/2ab(sinC). It can be simplified to the following form: h = a(sin C) to remove one unknown variable.

Now it remains to solve the resulting equation. For example, let "a" = 3, "C" = 40 degrees. Then the equation will look like this: "h" = 3(sin 40). Using a calculator and a sine table, calculate the value of "h". In our example, h = 1.928.