Find the height of the triangle using the radius of the inscribed circle. Equilateral triangle
Consider a circle inscribed in a triangle (Fig. 302). Recall that its center O is placed at the intersection of the bisectors of the interior angles of the triangle. The segments OA, OB, OS, connecting O with the vertices of the triangle ABC, will divide the triangle into three triangles:
AOB, BOS, SOA. The height of each of these triangles is equal to the radius, and therefore their areas are expressed as
The area of the whole triangle S is equal to the sum of these three areas:
where is the semiperimeter of the triangle. From here
The radius of the inscribed circle is equal to the ratio of the area of the triangle to its half-perimeter.
To obtain a formula for the radius of the circumscribed circle of a triangle, we prove the following proposition.
Theorem a: In any triangle, the side is equal to the diameter of the circumscribed circle multiplied by the sine of the opposite angle.
Proof. Consider an arbitrary triangle ABC and a circle circumscribed around it, the radius of which will be denoted by R (Fig. 303). Let A be the acute angle of the triangle. Let's draw the radii OB, OS of the circle and drop the perpendicular OK from its center O to the side BC of the triangle. Note that the angle a of a triangle is measured by half of the arc BC, for which the angle BOC is the central angle. From here it is clear that . Therefore, from a right-angled triangle SOK we find , or , which was required to be proved.
The given fig. 303 and the argument refer to the case of an acute angle of a triangle; it would not be difficult to carry out the proof for the cases of right and obtuse angles (the reader will do this on his own), but one can use the sine theorem (218.3). Since it must be where
The sine theorem is also written in. form
and comparison with the notation (218.3) gives for
The radius of the circumscribed circle is equal to the ratio of the product of the three sides of the triangle to its quadruple area.
A task. Find sides isosceles triangle, if its inscribed and circumscribed circles have radii, respectively
Solution. Let's write the formulas expressing the radii of the inscribed and circumscribed circles of the triangle:
For an isosceles triangle with a side and a base, the area is expressed by the formula
or, reducing the fraction by a non-zero factor , we have
that leads to quadratic equation relatively
It has two solutions:
Substituting instead of its expression into any of the equations for or R, we finally find two answers to our problem:
Exercises
1. The height of a right triangle, drawn from the vertex of the right angle, divides the hypotenuse in relation to Find the ratio of each of the legs to the hypotenuse.
2. Foundations isosceles trapezoid, circumscribed about a circle, are equal to a and b. Find the radius of the circle.
3. Two circles touch externally. Their common tangents are inclined to the line of centers at an angle of 30°. The length of the tangent segment between the points of contact is 108 cm. Find the radii of the circles.
4. The legs of a right triangle are equal to a and b. Find the area of a triangle whose sides are height and median given triangle drawn from the vertex of the right angle, and the segment of the hypotenuse between the points of their intersection with the hypotenuse.
5. The sides of the triangle are 13, 14, 15. Find the projection of each of them onto the other two.
6. In a triangle, the side and heights are known. Find the sides b and c.
7. Two sides of the triangle and the median are known. Find the third side of the triangle.
8. Given two sides of a triangle and an angle a between them: Find the radii of the inscribed and circumscribed circles.
9. The sides of the triangle a, b, c are known. What are the segments into which they are divided by the points of contact of the inscribed circle with the sides of the triangle?
A rhombus is a parallelogram with all sides equal. Therefore, it inherits all the properties of a parallelogram. Namely:
- The diagonals of a rhombus are mutually perpendicular.
- The diagonals of a rhombus are the bisectors of its interior angles.
A circle can be inscribed in a quadrilateral if and only if the sums of opposite sides are equal.
Therefore, a circle can be inscribed in any rhombus. The center of the inscribed circle coincides with the center of intersection of the diagonals of the rhombus.
The radius of an inscribed circle in a rhombus can be expressed in several ways
1 way. The radius of the inscribed circle in a rhombus through the height
The height of a rhombus is equal to the diameter of the inscribed circle. This follows from the property of a rectangle, which is formed by the diameter of the inscribed circle and the height of the rhombus - the opposite sides of the rectangle are equal.
Therefore, the formula for the radius of the inscribed circle in a rhombus through the height:
2 way. Radius of an inscribed circle in a rhombus through the diagonals
The area of a rhombus can be expressed in terms of the radius of the inscribed circle
, where R is the perimeter of the rhombus. Knowing that the perimeter is the sum of all the sides of a quadrilateral, we have P= 4×ha. Then
But the area of a rhombus is also half the product of its diagonals
Equating the right parts of the area formulas, we have the following equality
As a result, we obtain a formula that allows us to calculate the radius of the inscribed circle in a rhombus through the diagonals
An example of calculating the radius of a circle inscribed in a rhombus if the diagonals are known
Find the radius of a circle inscribed in a rhombus if it is known that the length of the diagonals is 30 cm and 40 cm
Let ABCD- rhombus, then AC and BD its diagonals. AC= 30 cm , BD=40 cm
Let the point O is the center of the inscribed in the rhombus ABCD circle, then it will also be the point of intersection of its diagonals, dividing them in half.
since the diagonals of the rhombus intersect at right angles, then the triangle AOB rectangular. Then by the Pythagorean theorem
, we substitute the previously obtained values into the formula
AB= 25 cm
Applying the previously derived formula for the radius of the circumscribed circle to a rhombus, we obtain
3 way. The radius of the inscribed circle in the rhombus through the segments m and n
Dot F- the point of contact of the circle with the side of the rhombus, which divides it into segments AF and bf. Let AF=m, BF=n.
Dot O- the center of intersection of the diagonals of the rhombus and the center of the circle inscribed in it.
Triangle AOB- rectangular, since the diagonals of the rhombus intersect at right angles.
, because is the radius drawn to the tangent point of the circle. Consequently OF- the height of the triangle AOB to the hypotenuse. Then AF and bf- projections of the legs onto the hypotenuse.
The height in a right triangle dropped to the hypotenuse is the average proportional between the projections of the legs on the hypotenuse.
The formula for the radius of an inscribed circle in a rhombus through the segments is equal to the square root of the product of these segments into which the side of the rhombus is divided by the tangent point of the circle
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If a circle is located inside an angle and touches its sides, it is called inscribed in this angle. The center of such an inscribed circle is located at bisector of this angle.
If it lies inside a convex polygon and is in contact with all its sides, it is called inscribed in a convex polygon.
Circle inscribed in a triangle
A circle inscribed in a triangle touches each side of this figure at only one point. Only one circle can be inscribed in one triangle.
The radius of such a circle will depend on the following parameters of the triangle:
- The length of the sides of a triangle.
- His area.
- Its perimeter.
- The angles of a triangle.
In order to calculate the radius of the inscribed circle in a triangle, it is not always necessary to know all the parameters listed above, since they are interconnected through trigonometric functions.
Calculation using the semi-perimeter
- If the lengths of all sides are known geometric figure(we denote them by the letters a, b and c), then you will have to calculate the radius by extracting the square root.
- Starting calculations, it is necessary to add one more variable to the initial data - the semi-perimeter (p). It can be calculated by adding all the lengths and dividing the resulting amount by 2. p = (a+b+c)/2. Thus, the formula for finding the radius can be significantly simplified.
- In general, the formula should include the sign of the radical under which the fraction is placed, the denominator of this fraction will be the value of the semi-perimeter p.
- The numerator of this fraction will be the product of the differences (p-a)*(p-b)*(p-c)
- Thus, the full form of the formula will be presented as follows: r = √(p-a)*(p-b)*(p-c)/p).
Calculation considering the area of a triangle
If we know area of a triangle and the lengths of all its sides, this will allow us to find the radius of the circle of interest to us without resorting to extracting roots.
- First you need to double the size of the area.
- The result is divided by the sum of the lengths of all sides. Then the formula will look like this: r = 2*S/(a+b+c).
- If you use the value of the semi-perimeter, you can get completely a simple formula: r = S/p.
Calculation using trigonometric functions
If the condition of the problem contains the length of one of the sides, the value of the opposite angle and the perimeter, you can use trigonometric function- tangent. In this case, the calculation formula will look like this:
r \u003d (P / 2- a) * tg (α / 2), where r is the desired radius, P is the perimeter, a is the length of one of the sides, α is the value of the opposite side, and the angle.
The radius of the circle to be inscribed in right triangle, can be found by the formula r = a*√3/6.
Circle inscribed in a right triangle
AT right triangle can be entered only one circle. The center of such a circle simultaneously serves as the intersection point of all bisectors. This geometric figure has some distinctive features, which must be taken into account when calculating the radius of the inscribed circle.
- First you need to build a right triangle with the given parameters. You can build such a figure by the size of its one side and the values of two angles, or by two sides and the angle between these sides. All these parameters must be specified in the task statement. A triangle is denoted as ABC, with C being the vertex of the right angle. The legs are denoted by variables, a and b, and the hypotenuse is a variable With.
- To build a classical formula and calculate the radius of a circle, it is necessary to find the dimensions of all sides of the figure described in the condition of the problem and calculate the semiperimeter from them. If the conditions give the dimensions of two legs, they can be used to calculate the value of the hypotenuse, based on the Pythagorean theorem.
- If the size of one leg and one angle is given in the condition, it is necessary to understand whether this angle is adjacent or opposite. In the first case, the hypotenuse is found using the sine theorem: с=a/sinСАВ, in the second case, the cosine theorem is applied с=a/cosCBA.
- When all calculations are completed and the dimensions of all sides are known, the semi-perimeter is found using the formula described above.
- Knowing the value of the semi-perimeter, you can find the radius. The formula is a fraction. Its numerator is the product of the differences of the semi-perimeter and each of the sides, and the denominator is the value of the semi-perimeter.
It should be noted that the numerator of this formula is an indicator of the area. In this case, the formula for finding the radius is much simpler - it is enough to divide the area by a half-perimeter.
It is also possible to determine the area of a geometric figure if both legs are known. The sum of the squares of these legs is the hypotenuse, then the semi-perimeter is calculated. You can calculate the area by multiplying the lengths of the legs by each other and dividing the result by 2.
If the conditions give the lengths of both the legs and the hypotenuse, you can determine the radius using a very simple formula: for this, the lengths of the legs are added, the length of the hypotenuse is subtracted from the resulting number. The result must be divided in half.
Video
From this video you will learn how to find the radius of a circle inscribed in a triangle.