Asked by Davis
In ΔXYZ, ∠X=52° and ∠Y=36°. ∠XWZ=90° and XY=10. Find the length of XW to the nearest 100th.
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Answered by
GPT-5 mini
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Let XZ = a. By the Law of Sines,
a = XZ = XY * sin(36°) / sin(92°) = 10 * sin36° / sin92°.
Place XY on the x-axis with X at the origin. The circle with diameter XZ meets the x-axis again at the projection of Z onto XY, so W is that projection and
XW = XZ · cos52° = a cos52°.
Thus
XW = 10 * (sin36° · cos52°) / sin92°.
Numerically,
sin36° ≈ 0.5877853, cos52° ≈ 0.6156615, sin92° ≈ 0.9993908,
so XW ≈ 10*(0.5877853*0.6156615)/0.9993908 ≈ 3.62096.
To the nearest hundredth, XW = 3.62.
a = XZ = XY * sin(36°) / sin(92°) = 10 * sin36° / sin92°.
Place XY on the x-axis with X at the origin. The circle with diameter XZ meets the x-axis again at the projection of Z onto XY, so W is that projection and
XW = XZ · cos52° = a cos52°.
Thus
XW = 10 * (sin36° · cos52°) / sin92°.
Numerically,
sin36° ≈ 0.5877853, cos52° ≈ 0.6156615, sin92° ≈ 0.9993908,
so XW ≈ 10*(0.5877853*0.6156615)/0.9993908 ≈ 3.62096.
To the nearest hundredth, XW = 3.62.
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