All about right triangles. How to find the sides of a right triangle? Basics of geometry
Properties right triangle
Dear seventh graders, you already know what geometric figures are called triangles, you know how to prove signs of their equality. You also know about special cases of triangles: isosceles and right angles. You are well aware of the properties of isosceles triangles.
But right triangles also have many properties. One obvious one is related to the theorem on the sum of interior angles of a triangle: in a right triangle, the sum sharp corners equals 90°. The most amazing property you will learn about a right triangle in 8th grade when you study the famous Pythagorean theorem.
Now we will talk about two more important properties. One is for 30° right triangles and the other is for random right triangles. Let us formulate and prove these properties.
You are well aware that in geometry it is customary to formulate statements that are converse to proven ones, when the condition and conclusion in the statement change places. Converse statements are not always true. In our case, both converse statements are true.
Property 1.1 In a right triangle, the leg opposite the 30° angle is equal to half the hypotenuse.
Proof: Consider the rectangular ∆ ABC, in which ÐA=90°, ÐB=30°, then ÐC=60°..gif" width="167" height="41">, therefore, what needed to be proved.
Property 1.2 (reverse to property 1.1) If in a right triangle the leg is equal to half the hypotenuse, then the angle opposite it is 30°.
Property 2.1 In a right triangle, the median drawn to the hypotenuse is equal to half the hypotenuse.
Let's consider a rectangular ∆ ABC, in which РВ=90°.
BD-median, that is, AD=DC. Let's prove that .
To prove this, we will make an additional construction: we will continue BD beyond point D so that BD=DN and connect N with A and C..gif" width="616" height="372 src=">
Given: ∆ABC, ÐC=90o, ÐA=30o, ÐBEC=60o, EC=7cm
1. ÐEBC=30o, since in a rectangular ∆BCE the sum of acute angles is 90o
2. BE=14cm(property 1)
3. ÐABE=30o, since ÐA+ÐABE=ÐBEC (property external corner triangle) therefore ∆AEB is isosceles AE=EB=14cm.
BC=2AN=20 cm (property 2).
Task 3. Prove that the altitude and median of a right triangle taken to the hypotenuse form an angle equal to the difference between the acute angles of the triangle.
Given: ∆ ABC, ÐBAC=90°, AM-median, AH-height.
Prove: RMAN=RS-RV.
Proof:
1)РМАС=РС (by property 2 ∆ AMC-isosceles, AM=SM)
2) ÐMAN = ÐMAS-ÐNAS = ÐS-ÐNAS.
It remains to prove that РНАС=РВ. This follows from the fact that ÐB+ÐC=90° (in ∆ ABC) and ÐNAS+ÐC=90° (from ∆ ANS).
So, RMAN = RС-РВ, which is what needed to be proven.
https://pandia.ru/text/80/358/images/image014_39.gif" width="194" height="184">Given: ∆ABC, ÐBAC=90°, AN-height, .
Find: РВ, РС.
Solution: Let's take the median AM. Let AN=x, then BC=4x and
VM=MS=AM=2x.
In a rectangular ∆AMN, the hypotenuse AM is 2 times larger than the leg AN, therefore ÐAMN=30°. Since VM=AM,
РВ=РВAM100%">
Let's extend AC beyond point A so that AD=AC. Then ∆ABC=∆ABD (on 2 legs). BD=BC=2AC=CD, thus ∆DBC-equilateral, ÐC=60o and ÐABC=30o.
Problem 5
In an isosceles triangle, one of the angles is 120°, the base is 10 cm. Find the height drawn to the side.
Solution: to begin with, we note that the angle of 120° can only be at the vertex of the triangle and that the height drawn to the side will fall on its continuation.
AB - staircase, K - kitten.
In any position of the ladder, until it finally falls to the ground, ∆ABC is rectangular. MC - median ∆ABC.
According to property 2 SK = 1/2AB. That is, at any moment in time the length of the segment SK is constant.
Answer: point K will move along a circular arc with center C and radius SC=1/2AB.
Problems for independent solution.
One of the angles of a right triangle is 60°, and the difference between the hypotenuse and the shorter leg is 4 cm. find the length of the hypotenuse. In a rectangular ∆ ABC with hypotenuse BC and angle B equal to 60°, the height AD is drawn. Find DC if DB=2cm. B ∆ABC ÐC=90o, CD - height, BC=2ВD. Prove that AD=3ВD. The altitude of a right triangle divides the hypotenuse into parts 3 cm and 9 cm. Find the angles of the triangle and the distance from the middle of the hypotenuse to the longer leg. The bisector splits the triangle into two isosceles triangles. Find the angles of the original triangle. The median splits the triangle into two isosceles triangles. Is it possible to find angles
The original triangle?
Side a can be identified as adjacent to angle B And opposite to angle A, and the side b- How adjacent to angle A And opposite to angle B.
Types of Right Triangles
- If the lengths of all three sides of a right triangle are integers, then the triangle is called Pythagorean triangle, and the lengths of its sides form the so-called Pythagorean triple.
Properties
Height
The height of a right triangle.
Trigonometric ratios
Let h And s (h>s) sides of two squares inscribed in a right triangle with a hypotenuse c. Then:
The perimeter of a right triangle is equal to the sum of the radii of the inscribed and three circumscribed circles.
Notes
Links
- Weisstein, Eric W. Right Triangle (English) on the Wolfram MathWorld website.
- Wentworth G.A. A Text-Book of Geometry. - Ginn & Co., 1895.
Wikimedia Foundation. 2010.
- Rectangular parallelepiped
- Direct costs
See what a “Right Triangle” is in other dictionaries:
right triangle- - Topics oil and gas industry EN right triangle ... Technical Translator's Guide
TRIANGLE- and (simple) trigon, triangle, man. 1. A geometric figure bounded by three mutually intersecting lines forming three internal corners(mat.). Obtuse triangle. Acute triangle. Right triangle... ... Dictionary Ushakova
RECTANGULAR- RECTANGULAR, rectangular, rectangular (geom.). Having a right angle (or right angles). Right triangle. Rectangular shapes. Ushakov's explanatory dictionary. D.N. Ushakov. 1935 1940 ... Ushakov's Explanatory Dictionary
Triangle- This term has other meanings, see Triangle (meanings). A triangle (in Euclidean space) is geometric figure, formed by three segments that connect three points that do not lie on the same straight line. Three dots,... ...Wikipedia
triangle- ▲ a polygon with three angles, a triangle, the simplest polygon; is defined by 3 points that do not lie on the same line. triangular. acute angle. acute-angled. right triangle: leg. hypotenuse. isosceles triangle. ▼… … Ideographic Dictionary of the Russian Language
TRIANGLE- TRIANGLE, huh, husband. 1. A geometric figure, a polygon with three angles, as well as any object or device of this shape. Rectangular t. Wooden t. (for drawing). Soldier's T. (soldier's letter without an envelope, folded in a corner; collapsible). 2... Ozhegov's Explanatory Dictionary
Triangle (polygon)- Triangles: 1 acute, rectangular and obtuse; 2 regular (equilateral) and isosceles; 3 bisectors; 4 medians and center of gravity; 5 heights; 6 orthocenter; 7 middle line. TRIANGLE, a polygon with 3 sides. Sometimes under... ... Illustrated Encyclopedic Dictionary
triangle Encyclopedic Dictionary
triangle- A; m. 1) a) A geometric figure bounded by three intersecting lines forming three internal angles. Rectangular, isosceles triangle. Calculate the area of the triangle. b) ott. what or with def. A figure or object of this shape... ... Dictionary of many expressions
Triangle- A; m. 1. A geometric figure bounded by three intersecting lines forming three internal angles. Rectangular, isosceles t. Calculate the area of the triangle. // what or with def. A figure or object of this shape. T. roofs. T.… … Encyclopedic Dictionary
Solving geometric problems requires a huge amount of knowledge. One of the fundamental definitions of this science is a right triangle.
This concept means consisting of three angles and
sides, with one of the angles measuring 90 degrees. The sides that make up a right angle are called legs, and the third side, which is opposite to it, is called the hypotenuse.
If the legs in such a figure are equal, it is called an isosceles right triangle. In this case, there is membership in two, which means that the properties of both groups are observed. Let us remember that the angles at the base of an isosceles triangle are absolutely always equal, therefore the acute angles of such a figure will include 45 degrees.
The presence of one of the following properties allows us to state that one right triangle is equal to another:
- the sides of two triangles are equal;
- the figures have the same hypotenuse and one of the legs;
- the hypotenuse and any of the acute angles are equal;
- the condition of equality of the leg and the acute angle is met.
The area of a right triangle is easily calculated both using standard formulas and as a value equal to half the product of its legs.
In a right triangle the following relations are observed:
- the leg is nothing more than the mean proportional to the hypotenuse and its projection onto it;
- if you describe a circle around a right triangle, its center will be in the middle of the hypotenuse;
- height drawn from right angle, represents the average proportional with the projections of the legs of the triangle onto its hypotenuse.
The interesting thing is that no matter what the right triangle is, these properties are always respected.
Pythagorean theorem
In addition to the above properties, right triangles are characterized by the following condition:
This theorem is named after its founder - the Pythagorean theorem. He discovered this relationship when he was studying the properties of squares built on
To prove the theorem, we construct a triangle ABC, the legs of which we denote as a and b, and the hypotenuse as c. Next we will build two squares. For one, the side will be the hypotenuse, for the other, the sum of two legs.
Then the area of the first square can be found in two ways: as the sum of the areas of four triangles ABC and the second square, or as the square of the side; naturally, these ratios will be equal. That is:
with 2 + 4 (ab/2) = (a + b) 2, we transform the resulting expression:
c 2 +2 ab = a 2 + b 2 + 2 ab
As a result, we get: c 2 = a 2 + b 2
Thus, the geometric figure of a right triangle corresponds not only to all the properties characteristic of triangles. The presence of a right angle leads to the fact that the figure has other unique relationships. Their study will be useful not only in science, but also in everyday life, since such a figure as a right triangle is found everywhere.
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 indicated 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 lowered from the vertex of a 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)