Math Problem Statement

AE is 1 + √3 long. AC is 1 long. BC is 1/√2 long. Find the shaded area.

Solution

To solve for the shaded area in the image, let's analyze the given information step by step:

Step 1: Understand the figure

The diagram shows a triangle ABE\triangle ABE with a right-angled triangle ABC\triangle ABC inside it, and a rectangle BCDEBCDE adjacent to it.

  • AE is the hypotenuse of triangle ABE\triangle ABE, and the length of AE is given as 1+31 + \sqrt{3}.
  • AC is one leg of the right-angled triangle ABC\triangle ABC, with length AC=1AC = 1.
  • BC is the other leg of the right-angled triangle, with BC=12BC = \frac{1}{\sqrt{2}}.

We need to find the area of the shaded rectangle BCDEBCDE.

Step 2: Calculate the area of the rectangle BCDEBCDE

The area of a rectangle is calculated using the formula: Area of the rectangle=length×width\text{Area of the rectangle} = \text{length} \times \text{width}

In this case:

  • The length of the rectangle is BC=12BC = \frac{1}{\sqrt{2}}.
  • The width of the rectangle is the vertical distance between points C and E, which corresponds to the height of triangle ABE\triangle ABE. To find CE, we need to calculate BEBE.

Step 3: Calculate the length of BE

Using the Pythagorean theorem in ABE\triangle ABE, we have: AE2=AB2+BE2AE^2 = AB^2 + BE^2

The length of AE is given as 1+31 + \sqrt{3}, and AB can be calculated as the sum of AC and BC: AB=AC+BC=1+12AB = AC + BC = 1 + \frac{1}{\sqrt{2}} So the equation becomes: (1+3)2=(1+12)2+BE2(1 + \sqrt{3})^2 = \left(1 + \frac{1}{\sqrt{2}}\right)^2 + BE^2 Expanding and solving for BE, we can then use that to find the vertical height of the rectangle (CE), and ultimately the area of the shaded region.

Let me complete the calculation steps for you.


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Math Problem Analysis

Mathematical Concepts

Geometry
Right-Angled Triangles
Area of Rectangle

Formulas

Area of Rectangle = length × width
Pythagorean Theorem

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 10-12