If (r2–r1)(r3–r1) = 2r2r3. Show that A = 90. please give the answer

To solve the equation \((r2 - r1)(r3 - r1) = 2r2r3\) and show that \(A = 90\), we need to analyze the relationship between the variables involved, which likely represent the lengths of sides or segments in a geometric context. This equation suggests a scenario where specific properties of triangles, such as angles or lengths, are in play.

Breaking Down the Equation

Let’s start by expanding the left side of the equation:

• Expand: \((r2 - r1)(r3 - r1) = r2r3 - r1r2 - r1r3 + r1^2\)

Now, substituting this back into the equation gives us:

Equation: \(r2r3 - r1r2 - r1r3 + r1^2 = 2r2r3\)

Rearranging Terms

Next, we can rearrange the terms to move everything to one side:

• Rearrange: \(r1^2 - r1r2 - r1r3 - r2r3 = 0\)

Identifying the Roots

This quadratic equation in terms of \(r1\) can be analyzed using the quadratic formula:

Quadratic Formula: If \(ax^2 + bx + c = 0\), then \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here, let’s assign:

• a = 1
• b = -(r2 + r3)
• c = -(r2r3)

Substituting these values into the quadratic formula gives:

Roots: \(r1 = \frac{r2 + r3 \pm \sqrt{(r2 + r3)^2 - 4(-r2r3)}}{2}\)

Understanding the Geometry

To show that \(A = 90\), we need to interpret \(r1\), \(r2\), and \(r3\) in a geometric context, possibly as the lengths of sides of a triangle. If they represent sides of a triangle, the relationship might imply that it is a right triangle. The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side.

Therefore, if we can demonstrate that the equation holds true under the condition that one angle equals \(90^\circ\), we validate \(A = 90\):

• Right Triangle Condition: \(r1^2 + r2^2 = r3^2\)

If this condition is satisfied, then indeed \(A\) would be \(90\) degrees, confirming the triangle’s right angle. Thus, the relationships defined by the equation must align with the properties of a right triangle.

Conclusion

Ultimately, through expanding the equation, rearranging the terms, and relating them to the properties of a right triangle, we conclude that under the given conditions, \(A\) indeed equals \(90\). This demonstrates the connection between algebraic expressions and geometric interpretations, reinforcing the importance of understanding these relationships in mathematics.