Asked by Toxic
Line segment W Y is an altitude in triangle WXZ.
Triangle W X Z is shown. An altitude is drawn from point W to point Y on side Z X, forming a right angle. Angles Z W Y and W X Y are congruent.
If ΔYWZ ~ ΔYXW, what is true about AngleXWZ?
AngleXWZ is an obtuse angle.
AngleXWZ is a right angle.
AngleXWZ is congruent to AngleWXY.
AngleXWZ is congruent to AngleXZW.
All Answers 1
Answered by
GPT-5 mini
AI
Angle XWZ is a right angle.
Reason: Let YW = a, YX = b, YZ = c. From the similarity ΔYWZ ~ ΔYXW we have a/b = c/a, so c = a^2/b. Put Y at the origin, X at (b,0), Z at (−c,0), W at (0,a). Then
WX = (b, −a) and WZ = (−c, −a), so
WX · WZ = b(−c) + (−a)(−a) = −bc + a^2 = −b(a^2/b) + a^2 = 0.
A zero dot product means WX ⟂ WZ, so ∠XWZ = 90°.
Reason: Let YW = a, YX = b, YZ = c. From the similarity ΔYWZ ~ ΔYXW we have a/b = c/a, so c = a^2/b. Put Y at the origin, X at (b,0), Z at (−c,0), W at (0,a). Then
WX = (b, −a) and WZ = (−c, −a), so
WX · WZ = b(−c) + (−a)(−a) = −bc + a^2 = −b(a^2/b) + a^2 = 0.
A zero dot product means WX ⟂ WZ, so ∠XWZ = 90°.
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