Question

determine the value of x. * 1 point

δabc ~ δxyz

your answer

Explanation:

Step1: Find the length of AB in ΔABC

In right triangle \( \triangle ABC \), \( AC \) is unknown, \( BC = 40 \), and \( AB = 50 \). Using the Pythagorean theorem \( AC=\sqrt{AB^{2}-BC^{2}}=\sqrt{50^{2}-40^{2}}=\sqrt{2500 - 1600}=\sqrt{900} = 30 \).

Step2: Set up the proportion for similar triangles

Since \( \triangle ABC \sim \triangle XYZ \), the ratios of corresponding sides are equal. The side \( AC = 30 \) in \( \triangle ABC \) corresponds to side \( YZ = 20 \) in \( \triangle XYZ \), and side \( AB = 50 \) in \( \triangle ABC \) corresponds to side \( XY = x \) in \( \triangle XYZ \). So the proportion is \( \frac{AC}{YZ}=\frac{AB}{XY} \), which is \( \frac{30}{20}=\frac{50}{x} \).

Step3: Solve for x

Cross - multiply: \( 30x=20\times50 \). Then \( 30x = 1000 \), and \( x=\frac{1000}{30}=\frac{100}{3}\approx33.33 \)? Wait, no, wait. Wait, maybe I mixed up the correspondence. Wait, \( \triangle ABC \sim \triangle XYZ \), so vertex \( A \) corresponds to \( X \), \( B \) to \( Y \), \( C \) to \( Z \). So \( BC \) corresponds to \( YZ \), \( AC \) corresponds to \( XZ \), \( AB \) corresponds to \( XY \). Wait, \( BC = 40 \), \( YZ = 20 \), so the scale factor is \( \frac{YZ}{BC}=\frac{20}{40}=\frac{1}{2} \). Then \( XY \) (which is \( x \)) corresponds to \( AB = 50 \), so \( x=50\times\frac{1}{2}=25 \)? Wait, no, my first step was wrong. Wait, \( \triangle ABC \): right - angled at \( C \), so \( AC \) and \( BC \) are legs, \( AB \) is hypotenuse. \( \triangle XYZ \): right - angled at \( Z \), so \( XZ \) and \( YZ \) are legs, \( XY \) is hypotenuse. So corresponding sides: \( AC \) (leg of \( \triangle ABC \)) corresponds to \( XZ \) (leg of \( \triangle XYZ \)), \( BC \) (leg of \( \triangle ABC \)) corresponds to \( YZ \) (leg of \( \triangle XYZ \)), \( AB \) (hypotenuse of \( \triangle ABC \)) corresponds to \( XY \) (hypotenuse of \( \triangle XYZ \)). So \( BC = 40 \), \( YZ = 20 \), so the ratio of similarity is \( \frac{YZ}{BC}=\frac{20}{40}=\frac{1}{2} \). Then the hypotenuse \( AB = 50 \) in \( \triangle ABC \) corresponds to hypotenuse \( XY=x \) in \( \triangle XYZ \), so \( x = 50\times\frac{1}{2}=25 \). Wait, but let's check with the legs. \( AC \) in \( \triangle ABC \): using Pythagoras, \( AC=\sqrt{50^{2}-40^{2}} = 30 \). Then \( XZ \) in \( \triangle XYZ \) should be \( 30\times\frac{1}{2}=15 \). Then check the legs of \( \triangle XYZ \): \( XZ = 15 \), \( YZ = 20 \), hypotenuse \( XY=\sqrt{15^{2}+20^{2}}=\sqrt{225 + 400}=\sqrt{625}=25 \). Yes, that works. So the correct proportion is based on the ratio of corresponding sides (hypotenuse to hypotenuse or leg to leg). Since \( BC = 40 \), \( YZ = 20 \), ratio \( \frac{YZ}{BC}=\frac{20}{40}=\frac{1}{2} \). Then \( AB = 50 \), so \( XY=x = 50\times\frac{1}{2}=25 \).

Answer:

\( \boxed{25} \)