When are the clock hands at a right angle between 8 and 9?

Question

At what time between 8:00 and 9:00 are the hour and minute hands exactly 90^\circ apart?

Accepted Answer

Use the angle formula. With H = 8, 30	imes 8 - 5.5M = 90 \quad\Longrightarrow\quad 240 - 5.5M = 90. This gives two cases. Case 1: 240 - 5.5M = 90 \Rightarrow 5.5M = 150 \Rightarrow M = 	frac{150}{5.5} = 	frac{300}{11} = 27	frac{3}{11} minutes. So one right angle is at 8\!:\!27	frac{3}{11}, about 8:27:16. Here the minute hand (near the 5-and-a-half mark) trails the hour hand (just past the 8) by exactly 90^\circ. Case 2: 240 - 5.5M = -90 \Rightarrow 5.5M = 330 \Rightarrow M = 60 minutes, i.e. 9:00 exactly. That is the boundary of the interval (the minute hand at 12, the hour hand at 9, a …

When are the clock hands at a right angle between 8 and 9?

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