Question

34 choose the appropriate value for x for the following similar triangles given below. △abc ~ △xyz. a y 20 z 50 x c x 40 b a 25 b 30 c 60 d 100

Explanation:

Step1: Identify corresponding sides

Since \(\triangle ABC \sim \triangle XYZ\), corresponding sides are proportional. In \(\triangle ABC\), \(BC = 40\), \(AB = 50\). In \(\triangle XYZ\), \(YZ = 20\), \(XY = x\). The ratio of \(BC\) to \(YZ\) should equal the ratio of \(AB\) to \(XY\).

Step2: Set up proportion

\(\frac{BC}{YZ}=\frac{AB}{XY}\) becomes \(\frac{40}{20}=\frac{50}{x}\). Simplify \(\frac{40}{20}=2\), so \(2=\frac{50}{x}\).

Step3: Solve for \(x\)

Cross - multiply: \(2x = 50\), then \(x=\frac{50}{2}=25\). Wait, no, wait. Wait, maybe I mixed up the correspondence. Wait, \(\triangle ABC\) has right angle at \(C\), \(\triangle XYZ\) has right angle at \(Z\). So \(BC\) (adjacent to right angle) in \(\triangle ABC\) corresponds to \(YZ\) (adjacent to right angle) in \(\triangle XYZ\), and \(AB\) (hypotenuse) in \(\triangle ABC\) corresponds to \(XY\) (hypotenuse) in \(\triangle XYZ\)? Wait, no, maybe \(BC\) corresponds to \(XZ\)? Wait, no, let's re - check. Let's find the length of \(AC\) first. In \(\triangle ABC\), using Pythagoras: \(AC=\sqrt{AB^{2}-BC^{2}}=\sqrt{50^{2}-40^{2}}=\sqrt{2500 - 1600}=\sqrt{900}=30\). Wait, maybe the correspondence is \(AC\) with \(XZ\), \(BC\) with \(YZ\), and \(AB\) with \(XY\)? Wait, no, the right angles are at \(C\) and \(Z\), so \(C\) corresponds to \(Z\), \(B\) corresponds to \(Y\), \(A\) corresponds to \(X\). So \(BC\) (side from \(B\) to \(C\)) corresponds to \(YZ\) (side from \(Y\) to \(Z\)), \(AC\) (side from \(A\) to \(C\)) corresponds to \(XZ\) (side from \(X\) to \(Z\)), and \(AB\) (side from \(A\) to \(B\)) corresponds to \(XY\) (side from \(X\) to \(Y\)). Wait, but \(YZ = 20\), \(BC = 40\), so the scale factor is \(\frac{YZ}{BC}=\frac{20}{40}=\frac{1}{2}\). Then \(XY\) (corresponding to \(AB\)) should be \(AB\times\frac{1}{2}\)? No, that gives \(25\), but wait, \(AC = 30\), maybe \(XZ\) is \(AC\times\frac{1}{2}=15\)? No, the options don't have 15. Wait, maybe I got the correspondence wrong. Let's try again. Let's assume that \(BC = 40\) corresponds to \(XY\) and \(AB = 50\) corresponds to \(YZ = 20\)? No, that would be \(\frac{40}{x}=\frac{50}{20}\), then \(50x = 800\), \(x = 16\), not an option. Wait, maybe the hypotenuse of \(\triangle ABC\) is \(AB = 50\), and the hypotenuse of \(\triangle XYZ\) is \(x\), and the leg \(BC = 40\) corresponds to leg \(YZ = 20\). So the ratio of legs is \(\frac{40}{20}=2\), so the ratio of hypotenuses should also be 2. So \(\frac{50}{x}=2\), so \(x = 25\)? But wait, \(AC\) is 30, maybe \(XZ\) is 15? No, the options are 25,30,60,100. Wait, maybe the correspondence is \(AC\) with \(YZ\) and \(BC\) with \(XZ\). Wait, \(AC = 30\), \(YZ = 20\)? No. Wait, maybe I made a mistake in Pythagoras. Wait, \(AB = 50\), \(BC = 40\), so \(AC=\sqrt{50^{2}-40^{2}}=\sqrt{2500 - 1600}=\sqrt{900}=30\). Now, if \(\triangle ABC\sim\triangle XYZ\), with right angles at \(C\) and \(Z\), then \(AC\) (leg) corresponds to \(XZ\) (leg), \(BC\) (leg) corresponds to \(YZ\) (leg), and \(AB\) (hypotenuse) corresponds to \(XY\) (hypotenuse). So the ratio of \(BC\) to \(YZ\) is \(\frac{40}{20}=2\), so the ratio of \(AC\) to \(XZ\) is also 2, so \(XZ=\frac{30}{2}=15\), and the ratio of \(AB\) to \(XY\) is 2, so \(XY=\frac{50}{2}=25\). But the option A is 25. Wait, but let's check the other way. If we take the ratio of \(YZ\) to \(AC\), \(20\) to \(30\) is \(\frac{2}{3}\), then \(BC\) is 40, so \(XZ\) would be \(40\times\frac{2}{3}\approx26.67\), not an option. So the first way, with \(BC\) corresponding to \(YZ\) (both legs adjacent to the right angle), and \(AB\) (hypotenuse) corresponding to \(XY\) (hypotenuse), the ratio of legs is 2, so hypotenuse ratio is 2, so \(x = 25\). But wait, the length of \(AC\) is 30, maybe the problem has a different correspondence. Wait, maybe \(AB\) corresponds to \(XZ\) and \(BC\) corresponds to \(YZ\). No, this is getting confusing. Wait, let's look at the options. If \(x = 25\), then the ratio of \(AB\) to \(XY\) is \(50/25 = 2\), and the ratio of \(BC\) to \(YZ\) is \(40/20=2\), so that's consistent. So the answer should be 25.

Answer:

A. 25