Pregunta
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Draw a trapezoid with two sides that are the same length.

Ask by Li Kelly. in the United States
Mar 09,2025

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To draw an isosceles trapezoid with two equal non-parallel sides, follow these steps: 1. **Draw the Bases:** - Draw a longer horizontal base \( AB \). - Draw a shorter horizontal base \( CD \) parallel to \( AB \). 2. **Mark the Vertices:** - Let \( A = (0,0) \), \( B = (4,0) \), \( D = (1,2) \), and \( C = (3,2) \). 3. **Connect the Points:** - Draw lines \( AB \), \( BC \), \( CD \), and \( DA \) to form the trapezoid. 4. **Verify Equal Sides:** - Calculate the lengths of \( AD \) and \( BC \): \[ AD = \sqrt{(1-0)^2 + (2-0)^2} = \sqrt{1+4} = \sqrt{5} \] \[ BC = \sqrt{(4-3)^2 + (0-2)^2} = \sqrt{1+4} = \sqrt{5} \] - Both \( AD \) and \( BC \) are equal, confirming the trapezoid is isosceles. 5. **Optional:** - Use TikZ in LaTeX to create a graphical representation of the trapezoid. This results in an isosceles trapezoid where the non-parallel sides are equal in length.

Solución

```markdown Below is one way to construct an isosceles trapezoid (a trapezoid with the two non-parallel sides equal in length): 1. **Draw the bases:** Choose a longer horizontal base \( AB \) and a shorter horizontal base \( CD \) that is parallel to \( AB \). 2. **Mark the vertices:** Let the vertices be: - \( A = (0,0) \) - \( B = (4,0) \) - \( D = (1,2) \) - \( C = (3,2) \) 3. **Connect the points:** Draw line segments to connect the points in order \( A \) to \( B \), \( B \) to \( C \), \( C \) to \( D \), and finally \( D \) to \( A \). 4. **Verify the equal sides:** Notice that the sides \( AD \) and \( BC \) are the non-parallel, slanted sides. Their lengths are equal if the following distances are the same: \[ AD = \sqrt{(1-0)^2 + (2-0)^2} = \sqrt{1+4} = \sqrt{5} \] \[ BC = \sqrt{(4-3)^2 + (0-2)^2} = \sqrt{1+4} = \sqrt{5} \] Hence, \( AD = BC = \sqrt{5} \). 5. **Graphical representation using TikZ (LaTeX):** ```latex \begin{tikzpicture}[scale=1] % Define the points \coordinate (A) at (0,0); \coordinate (B) at (4,0); \coordinate (C) at (3,2); \coordinate (D) at (1,2); % Draw the trapezoid \draw (A) -- (B) -- (C) -- (D) -- cycle; % Label the vertices \node[below left] at (A) {\(A\)}; \node[below right] at (B) {\(B\)}; \node[above right] at (C) {\(C\)}; \node[above left] at (D) {\(D\)}; % Optionally, indicate the equal non-parallel sides \node at ($(A)!0.5!(D)$) [left] {\(\sqrt{5}\)}; \node at ($(B)!0.5!(C)$) [right] {\(\sqrt{5}\)}; \end{tikzpicture} ``` The diagram above shows an isosceles trapezoid with bases \( AB \) and \( CD \) and with the non-parallel sides \( AD \) and \( BC \) equal in length. ```

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The Deep Dive

Did you know that trapezoids, or trapezia as they are sometimes called, have a fascinating place in geometry? The isosceles trapezoid, which has two sides of equal length, is not just a pretty shape; it showcases symmetry and balance. These types of trapezoids can be seen in architecture, like bridges and roof designs, highlighting their importance in both aesthetics and engineering efficiency. When drawing an isosceles trapezoid, avoid common mistakes like miscalculating the lengths of the non-parallel sides. A simple tip is to start by drawing the two parallel sides of equal length. Then, use a ruler to draw the equal-length sides at an angle so that they meet at the top. Don’t forget to check that the angles work out—getting it right ensures your shape is perfectly symmetrical!

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