Question

Solve for X and Y
X equals, Y equals

Answer 1

x = 3
`\sqrt6`

y = 3
`\sqrt{15`

Explanation:

We know that when a right triangle is split at its altitude, all three resulting triangles are similar.

This means that we can equate the ratios of their side lengths.

`\frac{\text{long leg of left triangle}}{\text{short leg of left triangle}} = \frac{\text{long leg of right triangle}}{\text{short leg of right triangle}}`

`(9)/(x) = (x)/(6)`

We can use this equation to solve for
`x`.

↓ multiplying both sides by
`x`

`9 = (x^2)/(6)`

↓ multiplying both sides by 6

`54 = x^2`

↓ taking the square root of both sides

`x = √(54)`

↓ simplifying the square root

`x=√(3^2 \cdot 6)`

`\boxed{x = 3\sqrt6}`

Now that we know what x is, we can solve for y using the Pythagorean Theorem.

`9^2 + x^2 = y^2`

↓ plugging in
`y`-value

`9^2 + √(54)^2 = y^2`

↓ simplifying exponents

`81 + 54 = y^2`

`y^2 = 135`

↓ taking the square root of both sides

`y=√(135)`

↓ simplifying the square root

`y=√(3^3 \cdot 5)`

`\boxed{y=3√(15)}`