NCERT Solutions

Construct a triangle with sidelengths 3 cm, 4 cm, and 8 cm. What is happening? Are you able to construct the triangle? Here is another set of lengths: 2 cm, 3 cm, and 6 cm. Check if a triangle is possible for these sidelengths.

Find examples of measurements of two angles with the included side where a triangle is not possible.

There is a spider in a corner of a box. It wants to reach the farthest opposite corner (marked in the figure). Since it cannot fly, it can reach the opposite point only by walking on the surfaces of the box. What is the shortest path it can take?

Find $\angle ACD$, if $\angle A=50°$, and $\angle B=60°$.

Find the exterior angle for different measures of $\angle A$ and $\angle B$. Do you see any relation between the exterior angle and these two angles? [Hint: From angle sum property, we have $\angle A+\angle B+\angle ACB=180°$. We also have $\angle ACD+\angle ACB=180°$, since they form a straight angle. What does this show?]

Construct a triangle ABC with BC=5 cm, AB=6 cm, CA=5 cm. Construct an altitude from A to BC.

Construct a triangle TRY with RY=4 cm, TR=7 cm, $\angle R=140°$. Construct an altitude from T to RY.

Construct a right-angled triangle $\triangle ABC$ with $\angle B=90°$, AC=5 cm. How many different triangles exist with these measurements?

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.

Can you construct a triangle all of whose angles are equal to 70°? If two of the angles are 70° what would the third angle be? If all the angles in a triangle have to be equal, then what must its measure be? Explore and find out.

Here is a triangle in which we know $\angle B=\angle C$ and $\angle A=50°$. Can you find $\angle B$ and $\angle C$?

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

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

Does there exist an equilateral triangle with sides 50, 50, 50? In general, does there exist an equilateral triangle of any sidelength? Justify your answer.

We have seen that triangles do not exist for all sets of sidelengths. Is there a combination of measurements in the case of two sides and the included angle where a triangle is not possible? Justify your answer using what you observe during construction.

We checked by construction that there are no triangles having sidelengths 3 cm, 4 cm and 8 cm; and 2 cm, 3 cm and 6 cm. Check if you could have found this without trying to construct the triangle.

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: $10 < 15+30$ $15 < 10+30$ But this doesn't happen for the third length: $30 > 10+15$. 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 for different sets of lengths. 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.] Given three sidelengths, what do we need to compare to check for the existence of a triangle?