Using knowledge of planimetry for solving some systems of algebraic equations

Aim: Perfecting students' skills for transferring knowledge of planimetry (metric dependencies in a right triangle, triangle's area and cosine rule) for solving systems of quadratic equations with tree or two unknowns (9th and 10th grade).

Example 1. If and are positive numbers and then without finding the values for and find the value of the expression

Solution: If we have to solve this system it would be easy for anyone manage to find and but in this problem is to be found. The system can be written down in the form

Since and are positive, from the first equation and can be considered using a reverse Pythagorean theorem as the lengths respectively of the legs and the hypotenuse in a right triangle ABD (fig.1)

 Figure 1.

If we consider the second equation of the system we can reach an analogical conclusion и are the respective lengths of the legs and hypotenuse in a triangle BCD (fig.1).

From the third equation we can make the conclusion that y is a number which is a geometrical average of and , and using the reverse theorem for proportional line segments in a right triangle the consequence is that

Now we examine the expression , which can be represented as . Considering figure 1 but

In this way we find out that i.e.

For this problem the question could be to find .

Example 2. If and are positive numbers and

then without finding values for and , calculate the expression .

Solution: The given system we represent in the form

Since and , considering the second equation of the system and having in mind a reverse Pythagorean theorem the numbers and are the lengths respectively of the legs and hypotenuse in a right triangle AMC (fig.2).

If we consider the first equation the numbers , and are lengths of sides in a triangle AMB in which and using a reverse cosine rule. Analogically , and are lengths of sides in a triangle BMC where ( fig. 2).

 Figure 2.

Since , then using a reverse Pythagorean theorem we can conclude that ABC is a right triangle and ACB =

We will find the areas of the following three triangles: using which the area of ABC can be found.

This way we get , i.e. .

#### For individual work

Problem 1. If and are positive numbers and then without finding the values of and find the value of the expression

Problem 2. Find , if , , and

Problem 3. If , , does the system have a solution:

Hint: If are solutions to the system, than the geometric interpretation of the system is presented in figure 3. Use the inequality between the sides of a triangle i.e.

 Figure 3.

Problem 4. Calculate the value of the expression , if and

Hint: and , because if and , then . The given system we represent in the form

Look above at the solution of problem 1.

Problem 5. Solve the system .

Hint: It is not hard to prove that and are positive numbers and and are lengths of the legs and hypotenuse in a right triangle.