Answer:
19.25 cm²
Step-by-step explanation:
A rectangle is inscribed in an equilateral triangle, with its base aligning with the base of the triangle.
The base of the equilateral triangle is divided into three equal segments. The base (width) of the rectangle is equal to one of these segments. Given that the base of the triangle is 10 cm, the width (w) of the rectangle is 10/3 cm.
[tex]w=\dfrac{10}{3}\; \sf cm[/tex]
The height (h) of the rectangle is equal to the height of the congruent right triangles. To find the height of the congruent right triangles, we can use the tangent trigonometric ratio.
[tex]\boxed{\begin{array}{l}\underline{\textsf{Tangent trigonometric ratio}}\\\\\sf \tan(\theta)=\dfrac{O}{A}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{$O$ is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{$A$ is the side adjacent the angle.}\end{array}}[/tex]
In this case:
Substitute the values into the equation and solve for h:
[tex]\tan 60^{\circ}=\dfrac{h}{\frac{10}{3}}[/tex]
[tex]h=\dfrac{10}{3}\tan 60^{\circ}[/tex]
[tex]h=\dfrac{10}{3}\cdot \sqrt{3}[/tex]
[tex]h=\dfrac{10\sqrt{3}}{3}\; \sf cm[/tex]
Therefore, the height of the rectangle is (10√3)/3 cm.
To find the area of the rectangle, multiply its width by its height:
[tex]\text{Area}=\dfrac{10}{3} \cdot \dfrac{10\sqrt{3}}{3}[/tex]
[tex]\text{Area}=\dfrac{100\sqrt{3}}{9}[/tex]
[tex]\text{Area}=19.24500897...[/tex]
[tex]\text{Area}=19.25\; \sf cm^2\;(nearest\;hundredth)[/tex]
Therefore, the area of the rectangle is 19.25 cm² (rounded to the nearest tenth).
