Answer:
AE = 43.2 units
Perimeter = 229.2 units
Step-by-step explanation:
Let the side AE be 'x'.
Consider triangles AEB and ADC
Statements Reasons
1. ∠ ABE ≅ ∠ ACD Right angles are congruent.
2. ∠A ≅ ∠A Common angle
Therefore, the two triangles are similar by AA postulate.
Now, for similar triangles, the ratio of their corresponding sides are also proportional to each other. Therefore,
[tex]\frac{AE}{AD}=\frac{AB}{AC}=\frac{BE}{DC}\\\\\frac{AE}{AE+ED}=\frac{AC-BC}{AC}[/tex]
Now, plug in the given values and solve for 'x'. This gives,
[tex]\frac{x}{x+72}=\frac{88-55}{88}\\\\88x=33(x+72)\\\\88x=33x+2376\\\\88x-33x=2376\\\\55x=2376\\\\x=\frac{2376}{55}=43.2[/tex]
Therefore, AE = 43.2 units
Now, from right angled triangle ABE,
[tex](AB)^2+(BE)^2=(AE)^2...(Pythagoras\ Theorem)\\\\(33)^2+(BE)^2=(43.2)^2\\\\BE=\sqrt{(43.2)^2-(33)^2}=27.879[/tex]
Similarly from right angled triangle ACD,
[tex]CD=\sqrt{AD^2-AC^2}\\\\CD=\sqrt{(72+43.2)^2-88^2}=74.344[/tex]
Now, perimeter is the sum of all the sides of a figure. Therefore, the perimeter of BCDE is given as:
Perimeter = BE + ED + CD + BC
Perimeter = 27.879 + 72 + 74.344 + 55 = 229.223 ≈ 229.2 (Nearest tenth)
Therefore, the perimeter = 229.2 units
