Pregunta
In rhombus \( A B C D \), the diagonals \( \overline{A C} \) and \( \overline{B D} \) intersect at \( E \). If \( A E=5 \) and \( B E=12 \), what is the length of \( \overline{A B} \) ? 1) 7 2) 10 3) 13 4) 17
Ask by Hilton Tyler. in China
Jan 23,2025
Solución de tutoría real
Respuesta verificada por el tutor
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\( \overline{AB} \) 的长度是 13，选择 3)。
Solución
在菱形 \( A B C D \) 中，两个对角线 \( \overline{AC} \) 和 \( \overline{BD} \) 互相垂直且平分。已知 \( AE = 5 \) 和 \( BE = 12 \)，因此对角线的全长分别为：
\[
AC = 2 \times AE = 2 \times 5 = 10
\]
\[
BD = 2 \times BE = 2 \times 12 = 24
\]
菱形的边长 \( AB \) 可以通过直角三角形的勾股定理来计算，其中 \( AB \) 是斜边，半对角线 \( \frac{AC}{2} = 5 \) 和 \( \frac{BD}{2} = 12 \) 是两条直角边：
\[
AB^2 = 5^2 + 12^2 = 25 + 144 = 169
\]
\[
AB = \sqrt{169} = 13
\]
因此，\( \overline{AB} \) 的长度是 13。
**答案:** 3) 13
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Simplifique esta solución Mind Expander
To find the length of \( \overline{A B} \) in rhombus \( ABCD \), we first note that the diagonals intersect at right angles and bisect each other. Given \( AE = 5 \) and \( BE = 12 \), we can determine the lengths of the diagonals: - The full length of diagonal \( AC \) is \( AE + EC = 5 + 5 = 10 \). - The full length of diagonal \( BD \) is \( BE + ED = 12 + 12 = 24 \). Now we can use the Pythagorean theorem to find \( AB \): \[ AB^2 = AE^2 + BE^2 = 5^2 + 12^2 = 25 + 144 = 169 \] Taking the square root gives: \[ AB = \sqrt{169} = 13 \] Thus, the length of \( \overline{A B} \) is \( 13 \). Therefore, the answer is \( \boxed{13} \).
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