Question

solve for x

Explanation:

Step1: Identify similar triangles

The two triangles are similar (by AA similarity, as they are right triangles and share the same angle). So, their corresponding sides are proportional.
Let the sides of the first triangle be \( x \), \( 48 \) (hypotenuse?), and the second triangle have sides \( 25 \), \( 30 \), and hypotenuse \( \sqrt{25^2 + 30^2} \)? Wait, maybe the sides are \( x \) (height), \( 48 \) (hypotenuse), and the smaller triangle has height \( 25 \), hypotenuse \( 30 \)? Wait, no, maybe the legs. Wait, looking at the diagram (assuming the first triangle has leg \( x \), hypotenuse \( 48 \), and the smaller triangle has leg \( 25 \), hypotenuse \( 30 \)? No, maybe the sides are proportional: \( \frac{x}{25}=\frac{48}{30} \)? Wait, no, maybe the other way. Wait, let's re - examine. If the two triangles are similar, then the ratio of corresponding sides is equal. Let's assume the first triangle has a leg of length \( x \), and the smaller triangle has a leg of length \( 25 \), and the hypotenuses are \( 48 \) and \( 30 \) respectively? Wait, no, maybe the legs are \( x \) and \( 48 \) (no, 48 is too long). Wait, maybe the first triangle has height \( x \), base (let's say) and the smaller triangle has height \( 25 \), base (let's say) and the hypotenuses are \( 48 \) and \( 30 \). Wait, the correct proportion: if the two triangles are similar, then \( \frac{x}{25}=\frac{48}{30} \)? Wait, no, \( \frac{x}{48}=\frac{25}{30} \)? No, let's do it properly. Let's assume that the two triangles are similar, so the ratio of corresponding sides is equal. Let the sides of the larger triangle be \( x \) (vertical leg), and the hypotenuse is \( 48 \), and the smaller triangle has vertical leg \( 25 \), hypotenuse \( 30 \). Then the proportion is \( \frac{x}{25}=\frac{48}{30} \). Wait, solving for \( x \): \( x=\frac{48\times25}{30} \).

Step2: Calculate the value of \( x \)

First, simplify \( \frac{48\times25}{30} \). We can cancel out common factors. \( 48\div6 = 8 \), \( 30\div6 = 5 \), \( 25\div5 = 5 \). So \( \frac{48\times25}{30}=\frac{8\times5\times5}{5}= 40 \)? Wait, no, \( 48\times25 = 1200 \), \( 1200\div30 = 40 \). Wait, but let's check the other way. If the smaller triangle has legs \( 15 \) and \( 20 \)? Wait, no, the numbers given: the smaller triangle has a side of \( 25 \) and hypotenuse \( 30 \)? No, maybe the legs are \( 15 \) and \( 20 \), hypotenuse \( 25 \)? Wait, I think I made a mistake. Wait, the Pythagorean triple: \( 15 - 20 - 25 \) (since \( 15^2+20^2 = 225 + 400=625 = 25^2 \)) and \( 24 - 32 - 40 \) ( \( 24^2+32^2=576 + 1024 = 1600=40^2 \))? Wait, no, the diagram shows the smaller triangle with side \( 25 \) (vertical) and hypotenuse \( 30 \)? No, that can't be, because \( 25^2=625 \), \( 30^2 = 900 \), \( 900-625 = 275 \), not a perfect square. Wait, maybe the sides are \( 25 \) (vertical), \( 15 \) (horizontal), and hypotenuse \( 30 \)? No, \( 15^2 + 25^2=225 + 625 = 850
eq900 \). Wait, maybe the correct proportion is \( \frac{x}{48}=\frac{25}{30} \)? No, \( x=\frac{48\times25}{30}=40 \). Wait, that gives \( x = 40 \), which is one of the options.

Answer:

\( 40 \)