Answer:
x = 1
Step-by-step explanation:
A right triangle is a triangle with a (90) degree angle. A box around one of the angles of a triangle indicates that this angle has a (90) degree measure. All of the triangles in the given picture are right triangles, as all of them contain a (90) degree angle. More than just being right triangles, the triangles in the diagram also have red arcs indicate that the angles are congruent. The base angles converse theorem states that in a triangle with two congruent angles, the sides opposite the congruent angles are also congruent, thus forming an isosceles triangle. Therefore, the given triangles are isosceles right triangles. One property of isosceles right triangles are that their sides follow the ratio:
[tex]a\ :\ a\ :\ a\sqrt{2}[/tex]
Where the sides with a measurement of (a) are the ones opposite the congruent angles, and the side with a measure of ([tex]a\sqrt{2}[/tex]) are the sides opposite the right angle. Apply this to the given triangle, refer to the attached diagram for further clarification about side namings,
[tex]BC*\sqrt{2}=AB\\\\BC=\frac{AB}{\sqrt{2}}\\\\BC=\frac{4}{\sqrt{2}}[/tex]
[tex]BD*\sqrt{2}=BC\\\\BD=\frac{BD}{\sqrt{2}}\\\\BD=\frac{\frac{4}{\sqrt{2}}}{\sqrt{2}}\\\\BD = \frac{4}{\sqrt{2}*\sqrt{2}}\\\\BD=\frac{4}{2}=2[/tex]
[tex]BE*\sqrt{2}=BD\\\\BE=\frac{BD}{\sqrt{2}}\\\\BE=\frac{2}{\sqrt{2}}[/tex]
[tex]BF*\sqrt{2}=BE\\\\BF=\frac{BE}{\sqrt{2}}\\\\BF=\frac{\frac{2}{\sqrt{2}}}{\sqrt{2}}\\\\BF = \frac{2}{\sqrt{2}*\sqrt{2}}=\frac{2}{2}=1[/tex]
[tex]BF = x = 1[/tex]
