Answer:
[tex]\sf A\:\!F[/tex] = 77.3 cm
Step-by-step explanation:
[tex]\sf A\:\!F[/tex] is the diagonal of rectangle AEFB and divides it into two congruent right triangles, AEF and ABF. Therefore, [tex]\sf A\:\!F[/tex] is the hypotenuse of right triangle ABF. To find the length of [tex]\sf A\:\!F[/tex], we can use the Pythagorean Theorem. However, we first need to determine the length of BF.
BF is the hypotenuse of right triangle BCF, where BC = 60 cm and ∠FBC = 25°. Since BC is the side adjacent to angle FBC, and we need to find the hypotenuse BF, we can use the cosine trigonometric ratio:
[tex]\sf \cos FBC=\dfrac{BC}{BF} \\\\\\\cos 25^{\circ}=\dfrac{60}{BF} \\\\\\BF=\dfrac{60}{\cos 25^{\circ}}[/tex]
Now, use the Pythagorean Theorem to find the length of [tex]\sf A\:\!F[/tex]:
[tex]\sf A\:\!F^2=AB^2+BF^2 \\\\\\ A\:\!F^2=40^2+\left(\dfrac{60}{\cos 25^{\circ}}\right)^2 \\\\\\ A\:\!F=\sqrt{1600+\left(\dfrac{60}{\cos 25^{\circ}}\right)^2} \\\\\\ A\:\!F=77.3485241966... \\\\\\A\:\!F=77.3\; cm\;(3\;s.f.)[/tex]
So, the length of [tex]\sf A\:\!F[/tex] correct to 3 significant figures is:
[tex]\LARGE\boxed{\boxed{\sf 77.3\; cm}}[/tex]
