Math Problem Statement

The diagram shows a hexagon ABCDEF with various side lengths given, and you are tasked with calculating the value of x based on the area of the hexagon being 258 cm².

Solution

The image shows a hexagon ABCDEFABCDEF with different side lengths and some variables included. We're given:

  • AB=25cmAB = 25 \, \text{cm}
  • BC=(x+2)cmBC = (x + 2) \, \text{cm}
  • CD=8cmCD = 8 \, \text{cm}
  • EF=7cmEF = 7 \, \text{cm}
  • AF=(x+6)cmAF = (x + 6) \, \text{cm}
  • EF=7cmEF = 7 \, \text{cm}

We are also told that the area of the hexagon is 258cm2258 \, \text{cm}^2 and are asked to find the value of xx.

Approach to solve for xx:

  1. Divide the hexagon into simpler shapes: The hexagon can be divided into a rectangle ABCDABCD and two right-angled triangles (one on the left and one on the right side).

  2. Area of the rectangle: The rectangle ABCDABCD has dimensions:

    • Height = AB=25cmAB = 25 \, \text{cm}
    • Width = BC=(x+2)cmBC = (x + 2) \, \text{cm}

    So, the area of rectangle ABCDABCD is: Area of rectangle=Height×Width=25×(x+2)\text{Area of rectangle} = \text{Height} \times \text{Width} = 25 \times (x + 2)

  3. Area of triangle DEFDEF: This is a right-angled triangle with:

    • Base EF=7cmEF = 7 \, \text{cm}
    • Height CDEF=87=1cmCD - EF = 8 - 7 = 1 \, \text{cm}

    The area of triangle DEFDEF is: Area of triangle=12×Base×Height=12×7×1=3.5cm2\text{Area of triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 7 \times 1 = 3.5 \, \text{cm}^2

  4. Area of trapezium AFEAFE: Using the trapezium AFEAFE, we can calculate the area based on given data, subtracting any unnecessary elements.

Let's calculate the value of xx.The value of xx is approximately 8.18cm8.18 \, \text{cm}.

Would you like further details or clarification on the steps involved?

Here are 5 related questions for practice:

  1. How would the formula for the area change if EFEF had a different length?
  2. Can you calculate the perimeter of the hexagon for x=8.18x = 8.18?
  3. How does the shape of the hexagon influence the calculation of its area?
  4. What would happen to the total area if one side of the hexagon was increased by 5 cm?
  5. How would you find the height of the hexagon if it was irregular?

Tip: For irregular polygons, dividing the shape into simpler geometric forms can make area calculations easier.

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Area Calculation

Formulas

Area of a rectangle: A = length * width
Area of a triangle: A = 1/2 * base * height
Area of a hexagon by dividing it into simpler shapes

Theorems

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Suitable Grade Level

Grades 10-12