Question 1

Use the points on the circle and/or the centre to form isosceles triangles.

Accepted Answer

Select any two points on the circle and connect them to the centre of the circle.
Also, join these points to each other. This will form an isosceles triangle as the two radii are equal in length.

Question 2

Use the points on the circles and/or their centres to form isosceles and equilateral triangles. The circles are of the same size.

Accepted Answer

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)

Question 3

For each set of lengths seen so far, you might have noticed that in at least two of the comparisons, the direct length was less than the sum of the other two (if not, check again!).
For example, for the set of lengths 10 cm, 15 cm, and 30 cm, there are two comparisons where this happens:
10  10 + 15.

Accepted Answer

Do it yourself.
NCERT In-Text Questions (Page 154)
Will this always happen? That is, for any set of lengths, will there be at least two comparisons where the direct length is less than the sum of the other two? Explore different sets of lengths.
Solution:
(i) 5 mm, 10 mm, 20 mm.
There are two comparisons where this happens:
10  10 + 5
(ii) 12 cm, 20 cm and 40 cm.
There are two comparisons where this happens:
12  12 + 20
Further, for a given set of lengths, is it possible to identify which lengths will immediately be less than the sum of the other two, without calculations?
[Hint: Consider the direct lengths in the increasing order]
Solution:
Yes, it is possible to identify which lengths will immediately be less than the sum of the other two, if we take the direct lengths in increasing order.
Figure it Out (Page 156)

Question 4

Through construction, explore if it is possible to construct an equilateral triangle that is (i) right-angled, (ii) obtuse-angled. Also construct an isosceles triangle that is (i) right-angled, (ii) obtuse-angled.

Accepted Answer

An equilateral triangle that is right-angled and obtuse-angled is not possible because each angle of an equilateral triangle is always 60°.
An isosceles right-angled triangle has one angle of 90° and the other two angles of 45° each.
An isosceles obtuse-angled triangle can have one angle as 120° and the other two angles of 30° each.

A Tale of Three Intersecting Lines