Question

26. Show that the three points (a, a),(-a,-a), and (-a√13, a√13) are the vertices of an
equilateral triangle
27. Show that the points (0, -1), (2, 1), (0,3) and (-2, 1) are the corners of a square​

Answer 1

Question :

• Show that the three points (a, a),(-a,-a), and (-a√13, a√13) are the vertices of an
• equilateral triangle
• Show that the points (0, -1), (2, 1), (0,3) and (-2, 1) are the corners of a square

Solution :

Part 1

Let ,

A(a,a),B(−a,−a) and C(−√3a,√3a) be the given points.

Then, we have

AB= √(−a−a)² +(−a−a)²

= √4a²+4a²

=2√2a

Now BC =√ (−√3a+a)² +(√ 3a+a)²

BC= √a²(1−√3) +a²(√3+1)²

BC=a √(1−√3 )²+(1+√3)²

BC=a√1+3−2√3 +1+3+2√3

=a√8

=2√2a

Now AC= (−√3a−a)²+( √3a−a)²

AC= √a² (√3+1) ²+a² (√3 −1) ²

AC=a√( √3+1)² +( √3)−1)²

AC=a√ (3+1+2√3+3+1−2√3)

=a √8

=2√2a

Clearly, we have

AB=BC=AC

Hence, the triangle ABC formed by the given points is an equilateral triangle.

Similarly you can find of square .