For going to a city B from city A, there is a route via city C such that AC⊥CB, AC = 2 x 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

Solution:

Given, there is a route via city C for going to a city B from city A

Also, AC⊥CB

A 26 km highway is proposed to be constructed which directly connects the two cities A and B.

We have to find the distance that will be saved in reaching city B from city A after the construction of the highway.

So, △ABC is right triangle with C at right angle.

By pythagoras theorem,

AB² = AC² + BC²

(26)² = (2x)² + [2(x + 7)]²

676 = 4x² + 4(x + 7)²

By using algebraic identity,

(a + b)² = a² + 2ab + b²

Now, (x + 7)² = x² + 14x + 49

So, 676 = 4x² + 4(x² + 14x + 49)

Dividing by 4 on both sides,

169 = x² + x² + 14x + 49

169 = 2x² + 14x + 49

2x² + 14x + 49 - 169 = 0

2x² + 14x - 120 = 0

Dividing by 2,

x² + 7x - 60 = 0

x² + 12x - 5x - 60 = 0

x(x + 12) - 5(x + 12) = 0

(x - 5)(x + 12) = 0

Now, x + 12 = 0

x = -12

Also, x -5 = 0

x = 5

Since a negative value is not possible, x = -12 is neglected.

So, x = 5 km

Put the value of x to find AB and BC

Now, AB = 2(5) = 10 km

BC = 2(5 + 7) = 2(12) = 24 km

So, the distance travelled via city C is 10 + 24 = 34 km

The distance saved in reaching city B from city A after the construction of the highway = 34 - 26 = 8 km.

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

✦ Try This: For going to a city Q from city P, there is a route via city R such that PR⊥RQ, PR = 5 x km and RQ = 3(x + 4) km. It is proposed to construct a 16 km highway which directly connects the two cities P and Q. Find how much distance will be saved in reaching city Q from city P after the construction of the highway.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 6

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NCERT Exemplar Class 10 Maths Exercise 6.4 Problem 6

For going to a city B from city A, there is a route via city C such that AC⊥CB, AC = 2 x 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

Summary:

For going to a city B from city A, there is a route via city C such that AC⊥CB, AC = 2 x 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. The distance that will be saved in reaching city B from city A after the construction of the highway is 8 km

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