Chapter 7 A Tale of Three Intersecting Lines (Class 7 - Latest Maths NCERT (Ganita Prakash I) NCERT Solutions)

Question. Construct triangles having the following sidelengths (all the units are in cm):

(a) $4, 4, 6$

(b) $3, 4, 5$

(c) $1, 5, 5$

(d) $4, 6, 8$

(e) $3.5, 3.5, 3.5$

Question 1. Use the points on the circle and/or the centre to form isosceles triangles.

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.

Question 1. We checked by construction that there are no triangles having sidelengths $3\text{ cm}, 4\text{ cm}$ and $8\text{ cm}$; and $2\text{ cm}, 3\text{ cm}$ and $6\text{ cm}$. Check if you could have found this without trying to construct the triangle.

Question 2. Can we say anything about the existence of a triangle for each of the following sets of lengths?

(a) $10\text{ km}, 10\text{ km}$ and $25\text{ km}$

(b) $5\text{ mm}, 10\text{ mm}$ and $20\text{ mm}$

(c) $12\text{ cm}, 20\text{ cm}$ and $40\text{ cm}$

You would have realised that using a rough figure and comparing the direct path lengths with their corresponding roundabout path lengths is the same as comparing each length with the sum of the other two lengths. There are three such comparisons to be made.

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 $10\text{ cm}, 15\text{ cm}$ and $30\text{ cm}$, 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.

Question. Which of the following lengths can be the sidelengths of a triangle? Explain your answers. Note that for each set, the three lengths have the same unit of measure.

(a) $2, 2, 5$

(b) $3, 4, 6$

(c) $2, 4, 8$

(d) $5, 5, 8$

(e) $10, 20, 25$

(f) $10, 20, 35$

(g) $24, 26, 28$

Question 1. Check if a triangle exists for each of the following set of lengths:

(a) $1, 100, 100$

(b) $3, 6, 9$

(c) $1, 1, 5$

(d) $5, 10, 12$

Question 2. 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.

Question 3. For each of the following, give at least $5$ possible values for the third length so there exists a triangle having these as sidelengths (decimal values could also be chosen):

(a) $1, 100$

(b) $5, 5$

(c) $3, 7$

Question. Construct triangles for the following measurements where the angle is included between the sides:

(a) $3\text{ cm}, 75^\circ, 7\text{ cm}$

(b) $6\text{ cm}, 25^\circ, 3\text{ cm}$

(c) $3\text{ cm}, 120^\circ, 8\text{ cm}$

Question. 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.

Question. Construct triangles for the following measurements:

(a) $75^\circ, 5\text{ cm}, 75^\circ$

(b) $25^\circ, 3\text{ cm}, 60^\circ$

(c) $120^\circ, 6\text{ cm}, 30^\circ$

Question 1. For each of the following angles, find another angle for which a triangle is (a) possible, (b) not possible. Find at least two different angles for each category:

(a) $30^\circ$

(b) $70^\circ$

(c) $54^\circ$

(d) $144^\circ$

Question 2. Determine which of the following pairs can be the angles of a triangle and which cannot:

(a) $35^\circ, 150^\circ$

(b) $70^\circ, 30^\circ$

(c) $90^\circ, 85^\circ$

(d) $50^\circ, 150^\circ$

Question. Like the triangle inequality, can you form a rule that describes the two angles for which a triangle is possible?

Question 1. Find the third angle of a triangle (using a parallel line) when two of the angles are:

(a) $36^\circ, 72^\circ$

(b) $150^\circ, 15^\circ$

(c) $90^\circ, 30^\circ$

(d) $75^\circ, 45^\circ$

Question 2. Can you construct a triangle all of whose angles are equal to $70^\circ$? If two of the angles are $70^\circ$ 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.

Question 3. Here is a triangle in which we know $\angle B = \angle C$ and $\angle A = 50^\circ$. Can you find $\angle B$ and $\angle C$?

Question 1. Construct a triangle $ABC$ with $BC = 5\text{ cm}$, $AB = 6\text{ cm}$, $CA = 5\text{ cm}$. Construct an altitude from $A$ to $BC$.

Question 2. Construct a triangle $TRY$ with $RY = 4\text{ cm}$, $TR = 7\text{ cm}$, $\angle R = 140^\circ$. Construct an altitude from $T$ to $RY$.

Question 3. Construct a right-angled triangle $\Delta ABC$ with $\angle B = 90^\circ$, $AC = 5\text{ cm}$. How many different triangles exist with these measurements?

[Hint: Note that the other measurements can take any values. Take $AC$ as the base. What values can $\angle A$ and $\angle C$ take so that the other angle is $90^\circ$?]

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.