Mathematics is an amazing science. However, such a thought only comes when you understand it. To achieve this, it is necessary to solve problems and examples, draw diagrams and drawings, and prove theorems.
The way to understanding geometry lies through the solution of problems. An excellent example is the task in which to find the area of an isosceles triangle.
What is an isosceles triangle, and what is its difference from the others?
In order not to be frightened of the terms "height", "area", "base", "isosceles triangle" and others, it will be necessary to start with theoretical bases.
First about the triangle. It is a flat figure, which is formed from three points - vertices, in turn, connected by segments. If two of them are equal to each other, then the triangle becomes isosceles. These sides were called lateral, and the rest became the basis.
There is a special case of an isosceles triangle - equilateral, when the third side is equal to two lateral triangles.
They prove to be faithful helpers in solving problems that require finding the area of an isosceles triangle. Therefore, it is necessary to know and remember them.
- The first of them: the angles of an isosceles triangle, one side of which is a base, are always equal to each other.
- An important property is the property of additional constructions. The height, median, and bisectrix drawn to the unpaired side coincide.
- The same segments drawn from the angles at the base of the triangle are pairwise equal. This also often makes it easier to find a solution.
- Two equal angles in it always have a value less than 90º.
- And the last thing: inscribed and circumscribed circumferences are constructed so that their centers lie at the height to the base of the triangle, and thus the median and the bisectrix.
How is it possible to recognize an isosceles triangle in the problem?
If, in solving a problem, the question arises of how to find the area of an isosceles triangle, then first you need to understand that it belongs to this group. And this will help certain signs.
- Two corners or two sides of a triangle are equal.
- The bisector is also a median.
- The height of the triangle turns out to be a median or bisectrix.
- Two heights, medians or bisectors of a figure are equal.
The notation of the quantities adopted in the formulas
To simplify how to find the area of an isosceles triangle by formulas, its elements are replaced by letters.
Notation in formulas
The letter in formula
Attention! It is important not to confuse "a" with "A" and "in" with "B". These are different values.
Formulas that can be used in different tasks
The lengths of the sides are known, and it is required to find the area of an isosceles triangle.
In this case, both values must be squared. The number obtained from the change in the lateral side is multiplied by 4 and subtracted from the second. From the obtained difference, extract the square root. The length of the base is divided by 4. Two numbers multiply. If you write these actions in letters, you get the following formula:
Let it be written under number 1.
The sides of the isosceles triangle are found from the values of the sides. A formula that someone might seem easier than the first.
The first action is to find half the base. Then find the sum and difference of this number with the side. The last two values multiply and extract the square root. The last action is to multiply everything by half the bottom. The letter equality will look like this:
The way to find the area of an isosceles triangle, if the base and height to it are known.
One of the shortest formulas. In it, you need to multiply both the given values and divide them by 2. This is how it will be written:
The number of this formula is 3.
In the assignment, the sides of the triangle and the value of the angle lying between the base and the side are known.
Here, in order to find out what the area of an isosceles triangle will be, the formula will consist of several factors. The first of these is the sine of the angle. The second is equal to the product of the lateral side on the base. The third is fraction ½. General mathematical notation:
The ordinal number of the formula is 4.
In the problem are given: the side of an isosceles triangle and the angle lying between its lateral sides.
As in the previous case, the area is in three multipliers. The first is equal to the sine of the angle indicated in the condition. The second is the side square. And the latter is also half of one. As a result, the formula is written as:
A formula that allows us to find the area of an isosceles triangle if its base and the angle lying opposite it are known.
First you need to calculate the tangent of half the known angle. The resulting number is multiplied by 4. Square the length of the side, which is then divided by the previous value. Thus, we get the following formula:
The number of the last formula is 6.
Examples of tasks
The first problem: it is known that the base of an isosceles triangle is 10 cm, and its height is 5 cm. It is necessary to determine its area.
To solve it, it is logical to choose the formula numbered 3. In it everything is known. Substitute the numbers and count. It turns out that the area is 10 * 5 / 2. That is 25 cm 2.
The second problem: in the isosceles triangle are given the side and the base, which are equal to 5 and 8 cm respectively. Find its area.
The first way. By the formula number 1. When you square the bottom, you get the number 64, and the quadrate of the lateral side is 100. After subtracting from the second one, you get 36. From it, the root is perfectly extracted, which is equal to 6. The base divided by 4 is equal to 2. The total value is defined as the product of 2 and 6, that is 12. This is the answer: the required area is 12 cm 2.
The second way. By the formula number 2. Half of the base is 4. The sum of the side and the found number gives 9, and their difference is 1. After multiplying, we get 9. The extraction of the square root yields 3. And the last action, multiplication by 3, gives the same 12 cm 2.
Tip: How to love mathematics
Solving problems in geometry and determining how to find the area of an isosceles triangle, you can get invaluable experience. The more different versions of tasks are performed, the easier it is to find the answer in a new situation. Therefore, regular and independent execution of all tasks is the way to successful mastering of the material.