Math Problem Statement
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 with a right-angled triangle inside it, and a rectangle adjacent to it.
- AE is the hypotenuse of triangle , and the length of AE is given as .
- AC is one leg of the right-angled triangle , with length .
- BC is the other leg of the right-angled triangle, with .
We need to find the area of the shaded rectangle .
Step 2: Calculate the area of the rectangle
The area of a rectangle is calculated using the formula:
In this case:
- The length of the rectangle is .
- The width of the rectangle is the vertical distance between points C and E, which corresponds to the height of triangle . To find CE, we need to calculate .
Step 3: Calculate the length of BE
Using the Pythagorean theorem in , we have:
The length of AE is given as , and AB can be calculated as the sum of AC and BC: So the equation becomes: 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.
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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
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