The lengths of the sides of a triangle are \( 5 x, 28 \) and \( 9 x-8 \). The perimeter is 76 . Solve for \( x \) and determine what kind of triangle it is. \( x=2 \), Right \( x=20 \), lsosceles \( x=4 \), Equilateral \( x=4 \), lsosceles

To find \( x \), we first express the perimeter of the triangle in terms of \( x \): \[ 5x + 28 + (9x - 8) = 76 \] Simplifying this, we get: \[ 14x + 20 = 76 \] Subtracting 20 from both sides gives us: \[ 14x = 56 \quad \Rightarrow \quad x = 4 \] Now, substituting back to find the lengths: - Side 1: \( 5(4) = 20 \) - Side 2: \( 28 \) - Side 3: \( 9(4) - 8 = 28 \) Thus, the sides are 20, 28, and 28, which means it's an isosceles triangle since two sides are equal. Isn't math incredible? Not only do we get the solve for \( x \), but we also uncover the beauty of triangles – with their various forms and properties! Next up, how many triangles need to be around before they realize that they can become an isosceles unit just by having two equal sides? Such a simple change can lead to countless possibilities in geometry, and every fascinating property—from the angles to the side relations—has a story worth exploring.