Answer:
[tex]\textsf{ Height of Triangle} = \boxed{ 12} \textsf{m}[/tex]
[tex]\textsf{Side of the Square }= \boxed{12 } \textsf{m}[/tex]
[tex]\textsf{Unknown base of the Trapezoid} =\boxed{12 } \textsf{m}[/tex]
[tex]\textsf{Height of the Trapezoid} =\boxed{3 } \textsf{m}[/tex]
[tex]\textsf{Area of Triangle }= \boxed{ 30} \textsf{m}^2[/tex]
[tex]\textsf{Area of the Square} = \boxed{144 } \textsf{m}^2[/tex]
[tex]\textsf{Area of the Trapezoid }= \boxed{27 } \textsf{m}^2[/tex]
[tex]\textsf{Area of Composite figure} = \boxed{ 201} \textsf{m}^2[/tex]
Step-by-step explanation:
Let's break down the calculations step by step:
Height of the Triangle:
The height of the triangle can be found using the Pythagorean theorem:
[tex]\sf \textsf{Height} = \sqrt{\textsf{Hypotenuse}^2 - \textsf{Base}^2} [/tex]
Given:
- Hypotenuse = 13 meters
- Base = 5 meters
So,
[tex]\sf \textsf{Height} = \sqrt{(13 \, \textsf{m})^2 - (5 \, \textsf{m})^2} [/tex]
[tex]\sf \textsf{Height} = \sqrt{169 \, \textsf{m}^2 - 25 \, \textsf{m}^2} [/tex]
[tex]\sf \textsf{Height} = \sqrt{144 \, \textsf{m}^2} [/tex]
[tex]\sf \textsf{Height} = 12 \, \textsf{m} [/tex]
Side of the Square:
Given that the side of the square is equal to the height of the triangle:
[tex]\sf \textsf{Side} = 12 \, \textsf{m} [/tex]
Base of the Trapezoid:
Given that the base of the trapezoid is equal to the side of the square:
[tex]\sf \textsf{Base} = 12 \, \textsf{m} [/tex]
Height of the Trapezoid:
Given that the height of the trapezoid is equal to the difference between the total height and the side of the square:
[tex]\sf \textsf{Height} = 15 \, \textsf{m} - 12 \, \textsf{m} [/tex]
[tex]\sf \textsf{Height} = 3 \, \textsf{m} [/tex]
Area of the Triangle:
[tex]\sf \textsf{Area} = \dfrac{1}{2} \times \textsf{Base} \times \textsf{Height} [/tex]
[tex]\sf \textsf{Area} = \dfrac{1}{2} \times 5 \, \textsf{m} \times 12 \, \textsf{m} [/tex]
[tex]\sf \textsf{Area} = 30 \, \textsf{m}^2 [/tex]
Area of the Square:
[tex]\sf \textsf{Area} = \textsf{Side}^2 [/tex]
[tex]\sf \textsf{Area} = (12 \, \textsf{m})^2 [/tex]
[tex]\sf \textsf{Area} = 144 \, \textsf{m}^2 [/tex]
Area of the Trapezoid:
[tex]\sf \textsf{Area} = \dfrac{1}{2} \times (\textsf{Base}_1 + \textsf{Base}_2) \times \textsf{Height} [/tex]
[tex]\sf \textsf{Area} = \dfrac{1}{2} \times (6 \, \textsf{m} + 12 \, \textsf{m}) \times 3 \, \textsf{m} [/tex]
[tex]\sf \textsf{Area} = 27 \, \textsf{m}^2 [/tex]
Area of the Composite Figure:
[tex]\sf \textsf{Area} = \textsf{Area of Triangle} + \textsf{Area of Square} + \textsf{Area of Trapezoid} [/tex]
[tex]\sf \textsf{Area} = 30 \, \textsf{m}^2 + 144 \, \textsf{m}^2 + 27 \, \textsf{m}^2 [/tex]
[tex]\sf \textsf{Area} = 201 \, \textsf{m}^2 [/tex]
Therefore, the area of the composite figure is [tex] \boxed{201 \, \textsf{m}^2} [/tex].
