ABCD is a rectangle. AB = 14, BC = 9. Find ED, where AC is given as 22.
Understand the Problem
The question presents a rectangle ABCD with given side lengths AB and BC, as well as the diagonal length AC. The goal is to find the length of the line segment ED, where E is the intersection point of the diagonals of the rectangle. Since diagonals of rectangles bisect each other, length ED will be half the length of diagonals of the rectangle. However, the diagonal should be calculated using the sides AB and BC to verify if the provided length of AC is correct.
Answer
$ED = 11$
Answer for screen readers
$ED = 11$
Steps to Solve
- Calculate the length of the diagonal AC using the Pythagorean theorem
In a rectangle $ABCD$, $AB$ and $BC$ are the sides, and $AC$ is the diagonal. According to the Pythagorean theorem: $AC^2 = AB^2 + BC^2$. So, $AC = \sqrt{AB^2 + BC^2}$. We are given that $AB = 14$ and $BC = 9$. Therefore, $AC = \sqrt{14^2 + 9^2} = \sqrt{196 + 81} = \sqrt{277} \approx 16.64$. Since the question states that $AC = 22$, there is an inconsistency in the problem's givens, meaning the rectangle described in the prompt is not a rectangle according to standard definitions of Euclidean geometry (in which $\sqrt{14^2 + 9^2} != 22$). However, the problem can still be solved if we accept $AC=22$ as a given.
- Determine the length of ED
In a rectangle, the diagonals bisect each other, which means they intersect at their midpoints. Therefore, $ED$ is half the length of the diagonal $AC$. $ED = \frac{AC}{2} = \frac{22}{2} = 11$.
$ED = 11$
More Information
The diagonals of a rectangle are equal in length and bisect each other. Therefore, the length of $ED$ is half the length of the diagonal $AC$. The Pythagorean theorem was used to calculate the diagonal length but using the sides of the rectangle led to a different number from the one provided in the problem. The answer makes use of the provided diagonal length, $AC = 22$, instead of the calculated one.
Tips
A common mistake is to calculate the diagonal $AC$ using the Pythagorean theorem and then use that calculated value instead of the given value of $AC$ to find $ED$. Another potential mistake might be assuming that $ED$ is equal to one of the sides of the rectangle.
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