Question

solve for x.
18
14
x
x = ?

Explanation:

Step1: Identify similar triangles

The two right triangles are similar, so corresponding sides are proportional.

Step2: Set up proportion

Let the common segment be $h$. For the large triangle: $\frac{x}{18}=\frac{18}{\text{hypotenuse of small triangle}}$, and the hypotenuse of the small triangle is $\sqrt{14^2 + h^2}$, also $x^2 = 18^2 + h^2$. Alternatively, use geometric mean theorem: $x^2 = 18 \times (18+14)$

Step3: Calculate $x^2$

$x^2 = 18 \times 32 = 576$? No, correction: Geometric mean for altitude gives $18^2 = 14 \times (x^2 -14^2)$? No, correct: The large triangle has hypotenuse $18+14=32$. By geometric mean, $x^2 = 18 \times 32$ is wrong. Correct: For the right triangle, the leg $x$ satisfies $x^2 = 18 \times 32$ is incorrect. Actually, using the property that in a right triangle, if an altitude is drawn to the hypotenuse, then $a^2 = p \times c$ where $a$ is leg, $p$ is adjacent segment, $c$ is hypotenuse. So $x^2 = 18 \times (18+14) = 18 \times 32 = 576$? No, $18*32=576$, $\sqrt{576}=24$? Wait no, correction: The two smaller triangles are similar to the large triangle. So $\frac{x}{32}=\frac{18}{x}$, so $x^2=18*32=576$, $x=24$. Wait no, the segments are 18 and 14 on the hypotenuse. So the leg $x$ corresponds to the hypotenuse 32 and the segment 18. So $x^2=18*32=576$, $x=24$.

Wait, correct step:

Step1: Recognize geometric mean theorem

In a right triangle, if an altitude is drawn to the hypotenuse, each leg is the geometric mean of the hypotenuse and the adjacent segment.

Step2: Apply theorem to leg $x$

The hypotenuse length is $18+14=32$, and the segment adjacent to $x$ is 18. So:
$$x^2 = 18 \times 32$$

Step3: Compute $x$

$$x^2 = 576$$
$$x = \sqrt{576} = 24$$

Final Answer: 24

Step1: Use geometric mean theorem

Leg $x$: $x^2 = 18 \times (18+14)$

Step2: Calculate $x^2$

$x^2 = 18 \times 32 = 576$

Step3: Solve for $x$

$x = \sqrt{576} = 24$

Answer:

$\sqrt{400}$ or 20