The value of x is 6.
To solve for x in the given image, we can use the Pythagorean Theorem on the small triangle. The Pythagorean Theorem states that:
a² + b² = c²
where:
a and b are the lengths of the two legs of the triangle
c is the length of the hypotenuse of the triangle
In the given image, we are given the following:
a = 4
b = x + 3
Substituting these values into the Pythagorean Theorem, we get:
4² + (x + 3)² = c²
16 + x² + 6x + 9 = c²
Simplifying, we get:
x² + 6x + 25 = c²
We can now use this value of c² as the base of the upper triangle. We are given that the height of the upper triangle is 8. We can now use the Pythagorean Theorem on the upper triangle to solve for x:
x² = (c²) - 8²
x² = (x² + 6x + 25) - 8²
x² = 6x + 9
Subtracting 6x from both sides of the equation, we get:
x² - 6x = 9
Factoring the left side of the equation, we get:
(x - 3)(x - 3) = 9
Taking the square root of both sides of the equation, we get:
x - 3 = ±3
Adding 3 to both sides of the equation, we get:
x = 3 ± 3
Therefore, the two possible solutions for x are:
x = 6 or x = 0
Since x cannot be 0, the only valid solution is x = 6.