For going to city B from city A, there is a route via city C such that AC ⟂ CB, AC = 2x km and CB = 2(x + 7) km.

It is proposed to construct a 26 km highway which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction of the highway.

From figure,

Distance between two cities A and B = AB = 26 km

Since, AC ⟂ CB

∠A = 90°

By Pythagoras theorem,

Hypotenuse² = Perpendicular² + Base²

In triangle ABC,

⇒ AB² = AC² + CB²

⇒ 26² = (2x)² + [2(x + 7)]²

⇒ 676 = 4x² + 4(x + 7)²

⇒ 676 = 4x² + 4(x² + 49 + 14x)

⇒ 676 = 4x² + 4x² + 196 + 56x

⇒ 8x² + 196 + 56x - 676 = 0

⇒ 8x² + 56x - 480 = 0

⇒ 8(x² + 7x - 60) = 0

⇒ x² + 7x - 60 = 0

⇒ x² + 12x - 5x - 60 = 0

⇒ x(x + 12) - 5(x + 12) = 0

⇒ (x - 5)(x + 12) = 0

⇒ x = 5 or x = -12.

Since, length cannot be negative.

x = 5

Distance traveled via city C = 2x + 2(x + 7) = 2x + 2x + 14 = 4x + 14 = 4 × 5 + 14 = 20 + 14 = 34 km.

Difference in distance traveled via city C and after construction of highway = 34 - 26 = 8 km.

Hence, 8 km distance will be saved in reaching city B from city A after the construction of the highway.