Just a right triangle.

$x+y$ = $\sqrt{9^{2}+12^{2}}$

$x+y$ = $\sqrt{81+144}$

$x+y$ = $\sqrt{225}$

$x+y$ = $15$

Then..

$\frac{12}{15}$ = $\frac{z}{9}$

$z$ = $7.2$

So..

$x+y+z$ = $15+7.2$

$x+y+z$ = $\boxed{22.2}$

Pythagorean Theorem: $x+y = 15$

Area of whole figure = $\frac{1}{2}\cdot9\cdot12 = 54$ .

However we can calculate the area another way, which is

$\frac{1}{2}\cdot(x+y)\cdot z$

But the area is $54,$ so

$\frac{1}{2}\cdot(x+y)\cdot z = 54$

Subsituting the value of $x+y$ and solving the equation, we will get $z = 7.2$

So, $x + y + z = 22.2$

I'm used to brute force method since I'm lacking knowledge for shortcuts in these matters. However, this is simply mathematics that happens to be possible to solve.

By Pythagoras Theorem for all three triangles,

$x+y=\sqrt{81+144}=15 ---(1)$

$81=x^2+z^2$ , left triangle

$144=y^z+z^2$ , right triangle

hence,

$x=\sqrt{81-z^2}---(2)$

$y=\sqrt{144-z^2}---(3)$

by substituting (2) and (3) into (1),

$\sqrt{81-z^2}+\sqrt{144-z^2}=15$

$\sqrt{81-z^2}=15-\sqrt{144-z^2}$

Square both sides, and we get,

$81-z^2=225+144-z^2-30\sqrt{144-z^2}$

$\sqrt{144-z^2}=9.6$

$z^2=144-9.6^2$

$z=\sqrt{144-9.6^2}$

$z=7.2$

Hence,

$x+y+z=22.2$

I actually have thought of using ratio, but I didn't know how to use it appropriately

Those are the stpes {y2 + z2 = 122 : Pythagora's theorem x2 + z2 = 92 : Pythagora's theorem (y + x)2 = 122 + 92 : Pythagora's theorem x + y = 15 : Solve equation C by extracting the square root y2 - x2 = 63 : subtracting equations A and B (y - x)(y + x) = 63 : factoring the left term of equation E. y - x = 21/5 $x = 27/5$ , $y = 48/5$ and $z = 36/5$ and then add them to get 22.2

By Pythagoras's theorem (x+y)= sqrt(81+144) = 15. Also, area of triangle = 9 12/2 =54 = (x+y) z/2 = 15z/2. Thus, z=7.2. Thus, x+y+z= 15+7.2 = 22.2.

It is a problem based on right angle altitude theorem first of all we get the sum of the (x+y) by pythagoras theorem Then by right angle altitude theorem we get 12^2=15y 9^2=15x z^2=xy

9x12=108
then z x (X+Y)=108 then X+Y=108/z and 9²+12²=(X+Y)²=225 then X+Y=15 15=108/z so Z=108/15=7.2 X+Y+Z=15+7.2=22.2

once we get x+y=15 using Pythagoras theorem .. we can equate the two formula's of area which are [0.5 base height]={[s (s-a) (s-b)*(s-c)]^0.5}... using this we can get the height(z).

Equating the Area of the right angled triangle through base (x+y) with Area of the right angled triangle through base 9 ...

1/2 ( 9 * 12 ) = 1/2 ( z * 15 ) this implies z = 7.2

Using Pythagoras Theorem, (x+y)^2 = 9^2 + 12^2 this implies (x+y) = 15

Hence, x+y+z = 15 + 7.2 = 22.2

in the given fig x+y=15 x^2+z^2=81 144=z^2+y^2 equating z y^2-x^2=63 from 1st equation (15-x)^2-x^2=63 x=5.4 81=(5.4)^2+z^2 z=7.2 y=15-5.4 y=9.6 x+y+z=22.2

All the angles of all the triangles are easily know,they are 53 degree and 37 degree,hence by applying cos and sin we can easily find the values of x,y,z

i did it using the following: area of triangle=9X12/2=54 also by pythagoras, x+y=15 also area of triangle=1/2XzX15=54 z=7.2 sum=22.2

by applying Puthagoras theorem then BC = sqrt(AB^2+AC^2) == sqrt(9^2+12^2) = 15. We have AH = AB AC/BC = 9 12/15 = 7.2. Then x + y + z = BC + AH = 22.2

Applying Pythagorean Theorem (a^2+b^2=c^2):

(9^2) + (12^2) = (x+y)^2

225 = (x+y)^2

15 = x+y

Getting the equation for each smaller triangle:

(9^2) = *[(15-y)^2] + (z^2)

(12^2) = *[(15-x)^2] + (z^2)

*Keep in mind that x+y = 15, and x = 15-y, y = 15-x

---> Notice that both equations contain a (z^2) <---

Rewrite each equation so that each will be equal to (z^2):

(z^2) = (9^2) - [(15-y)^2]

(z^2) = (12^2) - [(15-x)^2]

Rewrite into one equation:

(9^2) - [(15-y)^2] = (12^2) - [(15-x)^2]

Solve:

[(15-x)^2] - [(15-y)^2] = (12^2) - (9^2)

[225-30x+(x^2)]-[225-30y+(y^2)] = 144 - 81

-30x + (x^2) + 30y - (y^2) = 63

[(x^2)-(y^2)] - (30x+30y) = 63

*(x+y)(x-y) - 30(x-y) = 63

*Remember that x+y = 15

15(x-y) - 30(x-y) = 63

(x-y)(15-30) = 63

-15(x-y) = 63

x-y = -4.2

Now we have two equations:

x+y = 15 & x-y = -4.2

Solve:

x+y = 15

x-y = -4.2

2x = 10.8

x    =  5.4

(9^2) = (5.4^2) + (z^2)

81-29.16 = z^2

51.84 = z^2

7.2 = z

x+y+z = 15 + 7.2 = 22.2

Final Answer: 22.2