Question

In rectangle ABCD, AB = 3 and BC = 9. The rectangle is folded so that points A and

C coincide, forming the pentagon ABEFD. What is the length of segment EF? Express
your answer in simplest radical form.

Answer 1

Explanation:

To find the length of segment EF in pentagon ABEFD, we need to determine the distance between points E and F after the rectangle ABCD is folded.

Since AB = 3 and BC = 9, the length of the rectangle's diagonal, AC, can be found using the Pythagorean theorem. Let's denote the length of segment AC as x:

AC^2 = AB^2 + BC^2

x^2 = 3^2 + 9^2

x^2 = 9 + 81

x^2 = 90

x = sqrt(90)

x = 3sqrt(10)

When the rectangle is folded, points A and C coincide, meaning that the length of segment EF is equal to x.

Therefore, the length of segment EF in pentagon ABEFD is 3sqrt(10), expressed in simplest radical form.

Answer 2

The length of segment EF in pentagon ABEFD is 12.


Given:

Rectangle ABCD with AB = 3 and BC = 9

Folding the rectangle along AC to form pentagon ABEFD

Objective:

Determine the length of EF in pentagon ABEFD, in simplest radical form.

Solution:

Visualize the fold: Imagine folding rectangle ABCD along diagonal AC until points A and C coincide. This creates pentagon ABEFD with EF perpendicular to AD.

Identify similar triangles: Notice triangles ACB and AEF are similar due to sharing two corresponding angles (right angles and angle CAB).

Set up proportions: Use the ratio of corresponding sides in the similar triangles:

AC / EF = AB / CB

Substitute known values:

(AB + BC) / EF = AB / CB

(3 + 9) / EF = 3 / 9

12 / EF = 1/3

Solve for EF:

EF = 12 * 3

EF = 36 / 3

EF = 12

Final Answer:

The length of segment EF in pentagon ABEFD is 12.



The probable question is in the image attached.