Question

the figure shown is a rhombus. segment $ce$ has length 56, and $ab$ has length 106. find the length of $ae$.

Explanation:

Step1: Recall rhombus diagonals property

In a rhombus, diagonals bisect each other at right angles. So \( AE = DE \), \( CE = BE = 56 \), and \( \triangle AEB \) is a right triangle with \( AB = 106 \), \( BE = 56 \), and \( AE \) as the unknown leg.

Step2: Apply Pythagorean theorem

The Pythagorean theorem states that in a right triangle, \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse and \( a, b \) are the legs. Here, \( AB \) is the hypotenuse, \( BE \) is one leg, and \( AE \) is the other leg. So \( AE=\sqrt{AB^{2}-BE^{2}} \). Substitute \( AB = 106 \) and \( BE = 56 \):
\( AE=\sqrt{106^{2}-56^{2}}=\sqrt{(106 + 56)(106 - 56)}=\sqrt{162\times50}=\sqrt{8100}=90 \)

Answer:

\( 90 \)