Question

rectangle abcd is shown with 5 triangles inside it: abe, ade, aef, cde, and def.
a. which 2 triangles are similar to triangle aed?
b. the length of segment ef is 8 units and the length of segment fd is 6 units. find the length of segment fa.
type the answer in the box below.

Explanation:

Step1: Identify similar triangles

Since \(ABCD\) is a rectangle, \(\angle B=\angle C=\angle AFE = \angle DFE=90^{\circ}\).
For \(\triangle AED\) and \(\triangle ABE\):
\(\angle BAE+\angle EAD = 90^{\circ}\) and \(\angle ADE+\angle EAD=90^{\circ}\), so \(\angle BAE=\angle ADE\).
By AA (Angle - Angle) similarity criterion (\(\angle B=\angle AFE = 90^{\circ}\) and \(\angle BAE=\angle ADE\)), \(\triangle ABE\sim\triangle EFD\).
For \(\triangle AED\) and \(\triangle EFD\):
\(\angle DEF+\angle AEF = 90^{\circ}\) and \(\angle EAD+\angle AEF = 90^{\circ}\), so \(\angle DEF=\angle EAD\).
By AA (Angle - Angle) similarity criterion (\(\angle DFE=\angle AFE = 90^{\circ}\) and \(\angle DEF=\angle EAD\)), \(\triangle EFD\sim\triangle AED\).

Step2: Use the geometric mean theorem (for right - angled triangles)

In right - angled triangle \(AED\) with altitude \(EF\), we know that \(EF^{2}=FA\times FD\) (by the geometric mean theorem: in a right - triangle, the altitude drawn to the hypotenuse is the geometric mean between the segments of the hypotenuse).
Given \(EF = 8\) units and \(FD=6\) units.
Substitute into the formula \(8^{2}=FA\times6\).
So \(FA=\frac{64}{6}=\frac{32}{3}\) units.

Answer:

a. \(\triangle ABE\) and \(\triangle EFD\) are similar to \(\triangle AED\).
b. \(\frac{32}{3}\) units.