Question

In the diagram at right, triangle ABC~ triangle AED.

Answer 1

A. The fact that the question tells us that that two triangles are similar to each other, tells us that the corresponding sides of the triangles are proportionate to each other. This means that side AD is proportionate to AC and DE is proportionate to BC. This also means that the proportion that we get from AD and AC has to be equal to DE and BC (AD/AC has to equal AD/AC). With this, we can set up the following:

    x             10
-------- = ------------
  x+4         x+7

then, if we cross multiply, we get

x(x+7) = 10(x+4)

then after we simplify, we get

x^2 + 7x = 10x +40

then we solve it for zero and we get

x^2 -3x -40 =0 (the reason we solve for zero is to that we can factor it)

Then we factor

(x-8) = 0 and (x + 5) = 0

 and when we solve for x, we get

x =8 and x = -5, but since a negative number doesn't make sense, we find that x = 8 is the answer to part A. 

But this can easily be checked with the proportion (AD/AC has to equal AD/AC) 

   x             10
-------- = ------------
  x+4         x+7

so if we plug in 8 for x, we get

   8             10
-------- = ----------      and when simplified, we get 2/3 for both.
  12           15 

B. To figure of the perimeter, we must use the Pythagorean Theorem. 

a^2 + b^2 = c^2 

we will use a = 8 (this is what we got for x), b = 10 (this is given), and c for the hypotenuse.
Therefore, we get

8^2 + 10^2 = c^2

64 + 100 = c^2

164 = c^2
          __
c = 2 /41 (supposed to be 2 squareroot 41)
                                                                                                           __ 
Then you add 8,  10 and 2 root 41 to get your answer to be 18 + 2 /41 as the perimeter. 

This is the best that I've got.

Answer 2

Answer to part A: x = 8
Answer to part B: 30.80624847

The answer to part B is approximate. Round it however you need to

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Work Shown for part A

Because the triangles are similar, we can form the proportion shown below
ED/BC = AD/AC

Then using substitution, we can say
ED/BC = AD/AC
x/(x+4) = 10/(x+7)

Cross multiply and solve for x
x/(x+4) = 10/(x+7)
x(x+7) = 10(x+4)
x^2+7x = 10x+40
x^2+7x-10x-40 = 0
x^2-3x-40 = 0
(x-8)(x+5) = 0
x-8 = 0 or x+5 = 0
x = 8 or x = -5

Since ED = x, this means that x must be positive. Negative lengths don't make any sense.

So x = 8 is the only practical solution

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Work Shown for part B

Triangle AED is the smaller triangle with the legs
AD = 10
ED = x = 8

The hypotenuse is AE which is unknown for now. Let's call it y
AE = y

Use the pythagorean theorem to find the value of y

(AD)^2 + (ED)^2 = (AE)^2
10^2 + 8^2 = y^2
100 + 64 = y^2
164 = y^2
y^2 = 164
sqrt(y^2) = sqrt(164) ... "sqrt" stands for "square root"
y = sqrt(164)
y = 12.80624847 ... use a calculator to compute the square root of 164

Therefore AE is roughly 12.80624847 units long

Now that we know the three sides of triangle AED to be
AD = 10
ED = 8
AE = 12.80624847

Simply add up the three sides to get the perimeter

P = side1 + side2 + side3
P = AD + ED + AE
P = 10 + 8 + 12.80624847
P = 30.80624847

The perimeter is approximately 30.80624847
Round this value however you need to