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,
![\frac{AE}{AD}=\frac{AB}{AC}=\frac{BE}{DC}\\\\\frac{AE}{AE+ED}=\frac{AC-BC}{AC}](https://tex.z-dn.net/?f=%5Cfrac%7BAE%7D%7BAD%7D%3D%5Cfrac%7BAB%7D%7BAC%7D%3D%5Cfrac%7BBE%7D%7BDC%7D%5C%5C%5C%5C%5Cfrac%7BAE%7D%7BAE%2BED%7D%3D%5Cfrac%7BAC-BC%7D%7BAC%7D)
Now, plug in the given values and solve for 'x'. This gives,
![\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](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7Bx%2B72%7D%3D%5Cfrac%7B88-55%7D%7B88%7D%5C%5C%5C%5C88x%3D33%28x%2B72%29%5C%5C%5C%5C88x%3D33x%2B2376%5C%5C%5C%5C88x-33x%3D2376%5C%5C%5C%5C55x%3D2376%5C%5C%5C%5Cx%3D%5Cfrac%7B2376%7D%7B55%7D%3D43.2)
Therefore, AE = 43.2 units
Now, from right angled triangle ABE,
![(AB)^2+(BE)^2=(AE)^2...(Pythagoras\ Theorem)\\\\(33)^2+(BE)^2=(43.2)^2\\\\BE=\sqrt{(43.2)^2-(33)^2}=27.879](https://tex.z-dn.net/?f=%28AB%29%5E2%2B%28BE%29%5E2%3D%28AE%29%5E2...%28Pythagoras%5C%20Theorem%29%5C%5C%5C%5C%2833%29%5E2%2B%28BE%29%5E2%3D%2843.2%29%5E2%5C%5C%5C%5CBE%3D%5Csqrt%7B%2843.2%29%5E2-%2833%29%5E2%7D%3D27.879)
Similarly from right angled triangle ACD,
![CD=\sqrt{AD^2-AC^2}\\\\CD=\sqrt{(72+43.2)^2-88^2}=74.344](https://tex.z-dn.net/?f=CD%3D%5Csqrt%7BAD%5E2-AC%5E2%7D%5C%5C%5C%5CCD%3D%5Csqrt%7B%2872%2B43.2%29%5E2-88%5E2%7D%3D74.344)
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