In the diagram at right, triangle abc~ triangle aed.

a. solve for x

b. calculate the perimeter of triangle aed

explain as thoroughly as possible, i've no idea how toi already asked six times

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




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



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