A Tale of Three Intersecting Lines — Question 2

An isosceles triangle can be formed by connecting the intersecting points of two circles and the centres of either circle. Here, isosceles triangles are AXY and BXY. An equilateral triangle can be formed by connecting the centers of the 2 equal circles and one of their intersecting point. Here, triangle AXB or triangle AYB is an equilateral triangle. An equilateral triangle can be formed by connecting the centres of the 3 equal circles. Here, triangle ABC is an equilateral triangle. In addition, ∆PAB, ∆QBC, and ∆RAC are also equilateral triangles. Also, the equilateral triangle is a special case of an isosceles triangle. So, all these equilateral triangles are also isosceles. Are Triangles Possible for any Lengths? NCERT In-Text Questions (Page 151) Construct a triangle with sidelengths 3 cm, 4 cm, and 8 cm. What is happening? Are you able to construct the triangle? Solution: Since the arcs from the points A and B do not meet. So, we are not able to construct the triangle with sidelengths 3 cm, 4 cm, and 8 cm. Here is another set of lengths: 2 cm, 3 cm, and 6 cm. Check if a triangle is possible for these side lengths. Solution: The arcs from points A and B do not meet. So, a triangle is not possible for sidelengths 2 cm, 3 cm, and 6 cm. Triangle Inequality NCERT In-Text Questions (Page 153) Can we say anything about the existence of a triangle having sidelengths 3 cm, 3 cm, and 7 cm? Verify your answer by construction. Solution: Here, let us choose the direct path length AB = 7 cm. And, the round about path length = BC + CA = 3 cm + 3 cm = 6 cm. Since the direct path between two vertices is longer than the roundabout path via the third vertex. So, this triangle is not possible. Also, by construction existence of a triangle having side lengths 3 cm, 3 cm, and 7 cm is not possible because if we draw arcs from point A and B then they do not meet. “In the rough diagram given alongside, is it possible to assign lengths in a different order such that the direct paths are always coming out to be shorter than the roundabout paths? If this is possible, then a triangle might exist.” Solution: No Is such a rearrangement of lengths possible in the triangle? Solution: No Figure it Out (Page 154)