Question

given that the pair of triangles is similar, find the length x. (let a = 3 and b = 12.)

Explanation:

Step1: Identify similar triangles ratio

Since triangles are similar, corresponding sides are proportional. Let the small triangle's leg be \( a = 3 \), big triangle's leg be \( b = 12 \), and small triangle's other leg (let's say) is equal to big triangle's leg \( x \) proportionally? Wait, re - check: The small right triangle and big right triangle, with one angle common (the non - right angle), so the ratio of corresponding sides. Let the small triangle have legs \( a = 3 \) and the other leg (let's assume the vertical leg) and the big triangle has legs \( x \) (vertical) and \( b = 12 \) (horizontal). The ratio of similarity: Let the small triangle's horizontal leg be equal to the big triangle's vertical leg? Wait, maybe the small triangle's leg \( a = 3 \) corresponds to the big triangle's leg \( x \), and the small triangle's other leg (let's say the hypotenuse - adjacent? No, better: For similar right triangles, the ratio of legs should be equal. Wait, maybe the small triangle has a leg of length equal to \( a = 3 \) and the big triangle has a leg of length \( x \), and the other legs: small has a leg (let's say the horizontal) equal to the big's vertical? No, perhaps the small triangle's leg (the one with length equal to the big's \( x \) - corresponding) and the big's leg \( b = 12 \) correspond to the small's \( a = 3 \) and another side. Wait, maybe the ratio is \( \frac{x}{a}=\frac{b}{x} \)? No, that's for geometric mean. Wait, no, similar triangles: corresponding sides are proportional. Let's assume that the small triangle has legs \( a = 3 \) and (let's say) \( x \) (but no, the big triangle has legs \( x \) and \( b = 12 \)). Wait, maybe the small triangle's leg (the one marked with the same angle) is \( a = 3 \), and the big triangle's leg (same angle) is \( x \), and the other legs: small's other leg is equal to big's \( b = 12 \)? No, that doesn't make sense. Wait, maybe it's a case of similar right triangles where one leg of the small triangle is \( a = 3 \), one leg of the big triangle is \( b = 12 \), and the other legs are \( x \) (small) and \( x \) (big)? No, the problem says "find the length \( x \)". Wait, maybe the two triangles are similar, so the ratio of corresponding sides is equal. Let's suppose that the small triangle has a leg of length \( a = 3 \) and the big triangle has a leg of length \( x \), and the other legs: small has a leg (let's say the hypotenuse - adjacent) and big has a leg \( b = 12 \). Wait, maybe the correct proportion is \( \frac{x}{a}=\frac{b}{x} \) (geometric mean in similar right triangles, since when you have two similar right triangles, the leg of the big triangle is the geometric mean of the leg of the small triangle and the hypotenuse - but no, maybe the two triangles are such that one is a scaled version. Wait, let's think again. If the two triangles are similar, then the ratio of corresponding sides is equal. Let's assume that the small triangle has a leg of length \( a = 3 \) and the big triangle has a leg of length \( x \), and the other legs: small has a leg (let's say length \( x \)) and big has a leg of length \( b = 12 \). No, that's not. Wait, maybe the small triangle's leg (the one with length \( a = 3 \)) corresponds to the big triangle's leg (length \( x \)), and the small triangle's other leg (length equal to the big's \( b = 12 \)) corresponds to the big triangle's leg (length \( x \))? No, that would be \( \frac{3}{x}=\frac{x}{12} \), cross - multiply: \( x^{2}=3\times12 = 36 \), so \( x = 6 \). Ah, that makes sense! Because in similar right triangles, the leg of the triangle is the geometric mean of the segments into which it divides the hypotenuse, but here maybe the two triangles are similar, so the ratio of the legs gives a proportion where \( x \) is the geometric mean of \( a \) and \( b \). So \( \frac{x}{a}=\frac{b}{x} \), so \( x^{2}=a\times b \).

Step2: Substitute values and solve

Given \( a = 3 \) and \( b = 12 \), then \( x^{2}=3\times12=36 \). Taking the square root of both sides, \( x=\sqrt{36} = 6 \).

Answer:

\( 6 \)