Math Problem Statement

Find the perimeter of the square given that its side length is (x - 3) and its diagonal is (3x).

Solution

Let's analyze the given problem:

The image shows a square with a diagonal. We are asked to find the perimeter of the square. The side length of the square is represented by x3x - 3, and the length of the diagonal is given as 3x3x.

To find the perimeter, we need to first determine the side length of the square.

  1. Since we have the square's diagonal, we can use the Pythagorean Theorem. In a square, the diagonal splits it into two right-angled triangles. The relationship between the side length aa and the diagonal dd is:

d2=a2+a2d^2 = a^2 + a^2 d2=2a2d^2 = 2a^2 d=a2d = a\sqrt{2}

In this case:

  • a=x3a = x - 3 (the side length),
  • d=3xd = 3x (the diagonal).

Substitute into the equation: 3x=(x3)23x = (x - 3)\sqrt{2}

Now, let's solve for xx.


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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Pythagorean Theorem

Formulas

Pythagorean Theorem: a^2 + a^2 = d^2
Diagonal of a square: d = a√2
Perimeter of a square: P = 4a

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 8-10