Sometimes the question of how to find the area of an isosceles triangle, stands not only for the pupils or students, but in real, practical life.For example, during construction it is necessary to finish the facade of which is under roof.How to calculate the required amount of material?
often faced with similar tasks craftsmen who work with fabric or leather.After all, many of the details that have to carve out a master, have just the shape of an isosceles triangle.
So, there are some ways to help you find the area of an isosceles triangle.The first - the calculation of its base and height.
solutions we need to build for the visibility triangle MNP MN and the base height of PO.Now something completed in the drawing: from the point P to draw a line parallel to the ground, but from the point of M - the line parallel to the altitude.The point of intersection we call Q. To learn how to find the area of an isosceles triangle, one must consider the resulting quadrilateral MOPQ, in which the side of the triangle, we have MP is its diagonal.
We first prove that it is a rectangle.Since we built it ourselves, we know that the parties MO and OQ are parallel.And the part of QM and OP also parallel.Angle POM direct means and the angle OPQ also direct.Consequently, the resulting chёtyrёhugolnik is a rectangle.Find the area is not difficult, it is the product of PO in the OM.OM - it is half the base of the triangle MPN.It follows that the area of a rectangle is constructed by us poluproizvedeniyu height of a right triangle on its base.
second step task ahead of us, how to determine the area of a triangle is proof of the fact that we received a rectangle over the area corresponds to a given isosceles triangle, that is, that the area of the triangle is also poluproizvedeniyu base and height.
compare to start triangle PON and PMQ.Both are rectangular, as right angle in one of them is formed by the height and angle of the line in the other corner is a rectangle.They are hypotenuse sides of an isosceles triangle, thus also equal.Catete the PO and QM are equal both parallel sides of the rectangle.Hence, the area of the triangle PON, and triangle PMQ equal.
QPOM area of a rectangle is equal to the area of the triangle PQM and MOP in total.Replacing heightened triangle triangle QPM PON, we obtain the sum given to us for the conclusion of the theorem triangle.Now we know how to find the area of an isosceles triangle at the base and height - to calculate their poluproizvedenie.
But you can learn how to find the area of an isosceles triangle on the bottom and side.Here, too, there are two options: the Pythagorean theorem and Gerona.Consider the solution using the Pythagorean theorem.For example, take the same isosceles triangle PMN with a height of PO.
In a right triangle POM MP - hypotenuse.Its square is equal to the sum of the squares of the PO and OM.Since OM - half of the base, which as we know, we could easily find and build a number of OM in the square.Subtracting from the square of the hypotenuse of that number, we find out what is the other leg of the square, which is the height of an equilateral triangle.Finding the square root of the height difference, and knew the right triangle, you can give the answer to the task before us.
You simply multiply the height of the base and divide in half.Why should do so, we have explained in the first embodiment of the evidence.
Sometimes you need to perform calculations on the side and corner.Then we find the height and base, using the formula of sine and cosine, and, again, they multiply and divide by two.
