A Tale of Three Intersecting Lines

Cut out a paper triangle. Fix one of the sides as the base. Fold it in such a way that the resulting crease is an altitude from the top vertex to the base. Justify why the crease formed should be perpendicular to the base.

Shortest Path in a Box!

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?

Take a cardboard box and mark the path that you think is the shortest from one corner to its opposite corner. Compare the length of this path with that of the paths made by your friends.

Hint:

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.

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.

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

(a) 10 km, 10 km and 25 km (b) 5 mm, 10 mm and 20 mm (c) 12 cm, 20 cm and 40 cm

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

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

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.

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

See if you can describe all possible lengths of the third side in each case, so that a triangle exists with those sidelengths. For example, in case (a), all numbers strictly between 99 and 101 would be possible.

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

(a) 3 cm, 75°, 7 cm

(b) 6 cm, 25°, 3 cm

(c) 3 cm, 120°, 8 cm

Construct triangles for the following measurements:

(a) 75°, 5 cm, 75°

(b) 25°, 3 cm, 60°

(c) 120°, 6 cm, 30°

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) $144^\circ$

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

(a) 35°, 150°

(b) 70°, 30°

(c) 90°, 85°

(d) 50°, 150°

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

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.

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

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, ∠R = 140°. Construct an altitude from T to RY.

Construct a right-angled triangle $\Delta$ABC with $\angle$B = 90°, AC = 5 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°?]

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.

What happens when the three vertices lie on a straight line?

Construct a triangle in which all the sides are of length 4 cm.

Context: Construct a triangle in which all the sides are of length 4 cm.

Q. How did you construct this triangle and what tools did you use? Can this construction be done only using a marked ruler (and a pencil)?

Q. How do we make this construction more efficient?

Context: Step 2: Construct another arc of radius $4\text{ cm}$ from $B$. Let $C$ be the point of intersection of the arcs.

Q. The construction ensures that both $AC$ and $BC$ are of length $4\text{ cm}$. Can you see why?

1. How do we construct triangles that are not equilateral?

2. Construct a triangle of sidelength 4 cm, 5 cm and 6 cm.

Context: Construct a triangle of sidelength 4 cm, 5 cm and 6 cm.

Q. How do we construct this triangle more efficiently?

Construct

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

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.

Try to find more sets of lengths for which a triangle construction is impossible. See if you can find any pattern in them.

Context: Clearly, the direct straight-line path from the tent to the tree is shorter than the roundabout path via the pole. In fact, the direct straight-line path is the shortest possible path to the tree from the tent. Will the direct path between any two points be shorter than the roundabout path via a third point? Clearly, the answer is yes.

Q. Can this understanding be used to tell something about the existence of a triangle having sidelengths $10\text{ cm}$, $15\text{ cm}$ and $30\text{ cm}$?

Can we say anything about the existence of a triangle having sidelengths 3 cm, 3 cm and 7 cm? Verify your answer by construction.

In the rough diagram in Fig. 7.4, is it possible to assign lengths in a different order such 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.

Context: "In the rough diagram in Fig. 7.4, 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."

Q. Is such rearrangement of lengths possible in the triangle?

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?

Context: Does a triangle exist with sidelengths $4\text{ cm}$, $5\text{ cm}$ and $8\text{ cm}$? This satisfies the triangle inequality: $8 < 4 + 5 = 9$

Q. Why do we not need to check the other two sides?

Now, suppose that a circle of radius 5 cm is constructed, centred at B. Can you draw a rough diagram of the resulting figure?

Will triangles always exist when a set of lengths satisfies the triangle inequality? How can we be sure?

Q. Let us study each of these cases by finding the relation between the radii (the smaller two lengths) and AB (longest length).

Context: Case 2: Circles do not intersect internally

Q. For this case to happen, what should be the relation between the radii and AB?

Can we use this analysis to tell if a triangle exists when the lengths satisfy the triangle inequality?

How will the two circles turn out for a set of lengths that do not satisfy the triangle inequality? Find 3 examples of sets of lengths for which the circles:

(a) touch each other at a point,

(b) do not intersect.

Frame a complete procedure that can be used to check the existence of a triangle.

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.

Do triangles exist for every combination of two angles and their included side? Explore.

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

It is clear that if the line from B is "inclined" sufficiently to the right, then it will not meet the line $l$.

(a) Try to find a possible $\angle B$ (marked in the figure) for this to happen. (b) What could be smallest value of $\angle B$ for the lines to not meet?

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

Can the sum of the two angles be used for framing this rule?

Context: Let us take two angles, say $60^\circ$ and $70^\circ$, whose sum is less than $180^\circ$. Let the included side be 5 cm.

Q. What could the measure of the third angle be? Does this measure change if the base length is changed to some other value, say 7 cm? Construct and find out.

In general, once the two angles are fixed, does the third angle depend on the included sidelength? Try with different pairs of angles and lengths.

Try experimenting with different triangles to see if there is a relation between any two angles and the third one. To find this relation, what data will you keep track of and how will you organise the data you collect?

Context: Consider a triangle ABC with $\angle B = 50^\circ$ and $\angle C = 70^\circ$. Suppose we construct a line XY parallel to BC through vertex A.

Q. We can see new angles being formed here: $\angle XAB$, and $\angle YAC$. What are their values?

Angle Sum Property

What can we say about the sum of the angles of any triangle?

There is a convenient way of verifying the angle sum property by folding a triangular cut-out of a paper. Do you see how this shows that the sum of the angles in this triangle is $180^\circ$?

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 $\angle A + \angle B + \angle ACB = 180^\circ$.]

We also have $\angle ACD + \angle ACB = 180^\circ$, since they form a straight angle.

What does this show?

What would the altitude from A to BC be in this triangle?

Construct an arbitrary triangle. Label the vertices A, B, C taking BC to be the base.

Construct the altitude from A to BC,

Context: Construction of the Altitudes of a Triangle Construct an arbitrary triangle. Label the vertices $A$, $B$, $C$ taking $BC$ to be the base. Construct the altitude from $A$ to $BC$.

Q. Constructing the altitude using just a ruler is not accurate. To get a more precise angle of $90^\circ$, we use a set square along with a ruler.

Can you see how to do this?

Does there exist a triangle in which a side is also an altitude?

Visualise such a triangle and draw a rough diagram.

Context: In our study of triangles, we have encountered the following types of triangles; equilateral, isosceles, scalene and right-angled triangles.

Q. Did you spot any other type of triangle?

What are the other types of triangles based on angle measures?

What could an acute-angled triangle be? Can we define it as a triangle with one acute angle? Why not?