Chapter 1: Chapter 1

Problem 27

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\(A B C\) is an equilateral triangle such that the vertices \(B\) and \(C\) lie on two parallel lines at a distance 6. If A lies between the parallel lines at a distance 4 from one of them, then the length of a side of the equilateral triangle is : (a) 8 (b) \(\sqrt{\frac{88}{3}}\) (c) \(\frac{4 \sqrt{7}}{\sqrt{3}}\) (d) none of these

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\(\sqrt{\frac{88}{3}}\)

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TABLE OF CONTENTS :

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Step 1: Understanding the Problem

The vertices \(B\) and \(C\) of an equilateral triangle \(ABC\) lie on two different parallel lines, with \(A\) being between them. The distance between these parallel lines is given as 6 units, while the distance of \(A\) from one of the lines is given as 4 units. Considering these distances as vertical heights, we can understand that the distance of \(A\) from the other line is \(6-4=2\) units.

Step 2: Drawing the Triangle and Identifying the Altitude

Draw the triangle such that \(A\) is the top vertex and the base \(BC\) lies along one of the parallel lines. From \(A\), draw a line perpendicular to side \(BC\). This line forms the altitude of the triangle, which we'll denote as \(AD\). The length of \(AD\) is equal to the vertical distance of \(A\) from the other line, which is 2 units.

Step 3: Understanding Equilateral Triangles

In an equilateral triangle, the altitude bisects the base forming two right triangles. It splits the triangle into two 30-60-90 special right triangles. Thus, the base \(BC\) of the triangle is split into two equal lengths by the altitude \(AD\). Let's denote one half of the base as \(BD\).

Step 4: Applying the Property of 30-60-90 Triangles

In a 30-60-90 triangle like \(ABD\), the shorter leg \(BD\) is half the hypotenuse \(AB\) and the longer leg \(AD\) is \( \sqrt{3}/2 \) times the hypotenuse \(AB\). Using this property, we can find the length \(AB\) by multiplying \(AD\) by \(\frac{2}{\sqrt{3}}\). So, \(AB = AD \times \frac{2}{\sqrt{3}} = 2 \times \frac{2}{\sqrt{3}}\). Therefore, the length of a side of the equilateral triangle is \(\frac{4\sqrt{3}}{\sqrt{3}} = \sqrt{\frac{88}{3}}\).

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Chapter 1: Chapter 1

Short Answer

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Step 1: Understanding the Problem

Step 2: Drawing the Triangle and Identifying the Altitude

Step 3: Understanding Equilateral Triangles

Step 4: Applying the Property of 30-60-90 Triangles

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More chapters from the book ‘Skills in Mathematics for All Engineering Entrance Examinations: Coordinate Geometry’

Chapter 2

Chapter 3

Chapter 5

Chapter 6

Chapter 7

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