Question

Solve for both variables.

Answer 1

`\boxed{\quad x= (65)/(12)\quad }`

`\boxed{\quad y = (91)/(12) \quad }`

Explanation:

The smaller and larger triangles you see in the image are similar because

• in each triangle, one of the angles is 90°
• there is a common angle
• therefore the third angles are equal

By the AAA t(Angle-Angle-Angle) theorem the two triangles are similar

AAA Theorem
If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.

The smaller triangle has sides 5 and x

The larger triangle has corresponding sides 5 + 7 = 12 and x + y

By the similarity theorem


`(5)/(12) = (x)/(x+y)\\\\` Equation [1]

Consider the larger triangle which is a right triangle with legs 12 and 5

The hypotenuse of this triangle = x + y

By the Pythagorean theorem,

hypotenuse² = sum of squares of the other two sides

=> (x + y)² = 12² + 5² = 144 + 25 = 169

Therefore
x + y = √169 = 13

So we have x + y = 13

Plugging this value of x + y = 13 into equation [1] we get

`(5)/(12) = (x)/(13)\\\\`

Switching sides we get

`(x)/(13) =(5)/(12)\\\\\rightarrow x = 13 * (5)/(12)\\\rightarrow x= (65)/(12)\\\\`

Since x + y = 13, we can plug the value of x into this equation to get

`(65)/(12) + y = 13\\\\\\y = 13 - (65)/(12)\\\\y = 13 * (12)/(12) - (65)/(12)\\\\y = (156)/(12) - (65)/(12)\\\\\\y = (156-65)/(12)\\\\y = (91)/(12)\\\\`

Answer:

`\boxed{\quad x= (65)/(12)\quad }`

`\boxed{\quad y = (91)/(12) \quad }`

Just an additional note
Because of similarity, the ratio x/y must be equal to 5/7** but this would not be directly using the similarity ratios of triangles but it can be proved:

we have

`(x)/(x+y) = (5)/(12)`

Flipping both sides we get

`(x+y)/(x) = (12)/(5)\\\\(x)/(x) + (y)/(x) = (12)/(5)\\\\1 + (y)/(x) = (12)/(5)\\\\\\(y)/(x) = (12)/(5) - 1\\\\(y)/(x) = (7)/(5)\\\\Or,\\\\(x)/(y) = (5)/(7)`

and we can use this also to solve for x and y individually

**Incidentally, there is a theorem which directly provides this information. It is called the Side Splitter Theorem but not sure if you had heard about it so didn't want to go there