Two vertical poles of heights 11 m and 6 m stand 12 m apart on level ground. (i) Find the distance between their tops. (ii) A wire is tied from the top of the taller pole to a peg on the ground exactly midway between the poles. Find the length of the…

Answer: (i) 13 m (ii) \(\sqrt{157}\) m, about 12.5 m

[Figure: poles AB = 11 m and CD = 6 m standing on ground line BD, BD = 12 m; E on AB with CE parallel to BD; M midpoint of BD.]

(i) Draw CE \(\perp\) AB. Then BDCE is a rectangle, so CE = BD = 12 m and BE = CD = 6 m, giving AE = 11 − 6 = 5 m.

In right-angled \(\Delta AEC\): \(AC^2 = AE^2 + CE^2 = 25 + 144 = 169\) \(\therefore AC = 13\) m.

(ii) M is midway, so BM = 6 m. In right-angled \(\Delta ABM\): \(AM^2 = AB^2 + BM^2 = 121 + 36 = 157\) \(\therefore AM = \sqrt{157} \approx 12.53\) m.