Square and Triangle Geometry $$ x = 8 $$

Square and Triangle Geometry

In this lesson, we analyze a geometry problem involving a square and two right triangles. We are given the area of a shaded region and asked to find the value of x, which represents the side length of the square.

1. Theory of Area in Geometry

The area of a triangle can be calculated using the formula one half times the base times the height. By comparing the areas of different shapes, we can form equations that help us find unknown measurements.

2. Concept Explanation

Inside a square with side length x, there are two right triangles. The larger triangle uses the full side of the square as both its base and height. The smaller triangle inside has both its base and height equal to half of the square’s side. The shaded region represents the difference between the areas of these two triangles.

3. Steps or Formula

1. Write the area formula for a triangle.
2. Find the area of the large triangle.
3. Find the area of the small triangle.
4. Subtract the smaller area from the larger area.
5. Set the result equal to the given shaded area.
6. Solve for x.

4. Example Problem

Given:

The shaded green area is 24 square meters. The figure is a square with side length x. Inside the square are a large right triangle and a smaller right triangle.

Step 1: Find the area of the large triangle.

$$ A_{\text{large}} = \frac{1}{2} \cdot x \cdot x = \frac{x^2}{2} $$

Step 2: Find the area of the small triangle.

$$ A_{\text{small}} = \frac{1}{2} \cdot \frac{x}{2} \cdot \frac{x}{2} = \frac{x^2}{8} $$

Step 3: Write the equation for the shaded area.

$$ 24 = \frac{x^2}{2} - \frac{x^2}{8} $$

Step 4: Simplify the equation.

$$ 24 = \frac{4x^2 - x^2}{8} $$ $$ 24 = \frac{3x^2}{8} $$

Step 5: Solve for x.

$$ 3x^2 = 24 \times 8 $$ $$ 3x^2 = 192 $$ $$ x^2 = 64 $$ $$ x = 8 $$

Final Answer:

The value of x, the side length of the square, is 8 meters.