Math is a wonderful science. However, this idea only comes when you understand it. To achieve this, we need to solve problems and examples, draw diagrams and drawings to prove the theorems.
The path to understanding geometry is through problem solving. An excellent example is the job in which you need to find the area of an isosceles triangle.
What is an isosceles triangle, and how it is different from others?
To not be afraid of the terms "height", "area", "base", "isosceles triangle" and other need to start with the theoretical foundations.
First, the triangle. A flat figure, which consists of three vertex points, in turn, connected by line segments. If two of them are equal to each other, the triangle becomes isosceles. These parties have called side, and the remaining became the basis.
There is a private case of isosceles triangle — equilateral, when the third party is two side.
They are faithful assistants in tasks that require you to find the area of an isosceles triangle. So know and remember about them is necessary.
- The first of them: the angles of an isosceles triangle, one side of which — the base is always equal to each other.
- It is important and the property on additional constructions. Conducted by unpaired side height, median and angle bisector coincide.
- The same segments, drawn from the corners at the base of the triangle, are pairwise equal. This too often facilitates the search for a solution.
- Two equal angles it will always have a value less than 90º.
- And finally: the inscribed and circumscribed circles are constructed so that their centers lie at the height of the base of the triangle, which means the median and bisector.
As the task is to recognize an isosceles triangle?
If the solution of the task raises the question of how to find the area of an isosceles triangle, you must first understand that he belongs to this group. And this will help a certain characteristics.
- Equal to two angles or two sides of the triangle.
- The bisector is also a median.
- The altitude of the triangle is a median or angle bisector.
- Equal to two heights, medians or bisectors of the figure.
The values adopted in the formulas
To simplify how to find the area of an isosceles triangle using formulas, the replacement of its elements by letters is introduced.
Notation in the formulas
The letter in the formula
Attention! It is important not to confuse "a" with "A" and "b" with "B". This is a different value.
Formulas, which can be used in different tasks
Known the lengths of the sides, and you want to find the area of an isosceles triangle.
In this case, you need to square both values. The number that came from the lateral sides, multiply by 4 and subtract from it the second. From the obtained difference square root. The length of the base divided by 4. Two numbers to multiply. If you write these action letters, we get this formula:
Let it be recorded under number 1.
Find by the values of the sides the area of an isosceles triangle. A formula that may seem simpler to someone than the first.
The first step you need to find half of the base. Then find the sum and the difference of this number with the side. The last two values to multiply and take the square root. The last step is to multiply all on half Foundation. Literal equality would look like this:
A way to find the area of an isosceles triangle, if the base and height to it are known.
One of the shortest formulas. It is necessary to multiply both these values and divide them by 2. Here's how it will be written:
The number of this formula 3.
In the task, the sides of the triangle and the angle between the base and the side are known.
Here, in order to know, what is the area of an isosceles triangle, the formula will consist of several factors. The first of these is the value of the sine of the angle. The second is the product of the sides to the base. The third fraction is½. Common mathematical notation:
The sequence number of formula 4.
In the problem are given: the side of an isosceles triangle and the angle lying between its sides.
As in the previous case, the area is three multipliers. The first is equal to the value of the sine of the angle specified in the condition. The second is the square side. And the latter is also equal to half the unit. In the end, the formula can be written as:
The formula that allows you to find the area of an isosceles triangle, if you know its base and the angle lying opposite it.
First you need to calculate the tangent of half of the known angle. The resulting number is multiplied by 4. Squared length of the lateral side, which is then divided by the previous value. Thus, we obtain a formula:
The number of the last formula is 6.
Examples of tasks
The first task: 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.
For its solution it is logical to select a formula number 3. It all known. To substitute the numbers and count. Turns out that the area is equal to 10 * 5 / 2. That is 25 cm 2 .
The second task: in an isosceles triangle, the side and the base are given, which are respectively 5 and 8 cm. Find its area.
The first method. According to formula No. 1. With squared base, the number of turns 64 and quadruple the square side part — 100. After subtraction of the first from the second work 36. From it perfectly retrieves the root, which is equal to 6. The base is divided by 4, is 2. The final value will be determined as the product of 2 and 6, i.e. 12. This is the answer: the desired area is 12 cm 2 .
The second method. According to the formula No. 2. Half the base is equal to 4. The sum of the sides and found the number 9 gives the difference — 1. After multiplying the obtained 9. The square root gives 3. And the last step, the multiplication of 3 by 4, which gives the same 12 cm 2 .
Advice: how to love math
Solving problems in geometry and determining how to find the area of an isosceles triangle, you can get invaluable experience. The more various options of quests you complete, the easier it is to find the answer in a new situation. Therefore, regular and independent performance of all jobs is the way to successful learning.