A Tale of Three Intersecting Lines | FIO

Question 7

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.

Solution

Understand the Question

Triangle Inequality Theorem: A triangle can exist if and only if the sum of the lengths of any two sides is strictly greater than the length of the third side.
- $a + b > c$
- $a + c > b$
- $b + c > a$
For an equilateral triangle, all three sides are equal ( $a = b = c = x$ ). Since length must be positive ( $x > 0$ ), checking $x + x > x \implies 2x > x$ is sufficient to verify if it can exist.

(i) Does there exist an equilateral triangle with sides 50, 50, 50? Justify your answer.

Step 1 · Check Triangle Inequality for Sides 50, 50, 50

Diagram 1

Let the side lengths be $a = 50$ , $b = 50$ , and $c = 50$ .

Check the three triangle inequality conditions:

First condition:

\begin{aligned} a + b &> c \\ 50 + 50 &> 50 \\ 100 &> 50 \quad (\text{True}) \end{aligned}

Second condition:

\begin{aligned} a + c &> b \\ 50 + 50 &> 50 \\ 100 &> 50 \quad (\text{True}) \end{aligned}

Third condition:

\begin{aligned} b + c &> a \\ 50 + 50 &> 50 \\ 100 &> 50 \quad (\text{True}) \end{aligned}

Since the sum of any two sides ( $100$ ) is strictly greater than the third side ( $50$ ), the triangle exists.

Answer

(i) Yes, an equilateral triangle with sides $50, 50, 50$ exists.

(ii) In general, does there exist an equilateral triangle of any sidelength? Justify your answer.

Step 1 · Check Triangle Inequality for General Sidelength $x$

Let the sidelength of any equilateral triangle be $x$ , where $x > 0$ .

Applying the triangle inequality theorem:

\begin{aligned} x + x &> x \\ 2x &> x \end{aligned}

Since sidelength $x$ must be a positive real number ( $x > 0$ ), $2x > x$ is always true for any positive value of $x$ (e.g., if $x = 1$ , $2 > 1$ ; if $x = 0.5$ , $1 > 0.5$ ).

Therefore, an equilateral triangle exists for any positive sidelength.

Answer

(ii) Yes, an equilateral triangle exists for any positive sidelength $x > 0$ .

Common Mistakes

Ignoring Positive Length Constraint: Sidelengths must be strictly positive ( $x > 0$ ). A triangle cannot have a side length of zero or a negative value.
Checking Only Two Sides in Non-Equilateral Triangles: In general triangles, all three side pairings must be checked. For equilateral triangles, checking one pair ( $x + x > x$ ) verifies all three since all sides are equal.

More questions in FIO

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 $3\text{ cm}$ , $4\text{ cm}$ 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.

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 $20\text{ mm}$

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

(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) $5$ , $5$

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

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

Q10

Construct triangles for the following measurements:

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

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

Q11

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) $70^\circ$

(d) $144^\circ$

Q12

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

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

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

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

Q13

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

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

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

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

Q14

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.

Q15

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

Q16

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

Q17

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

Q18

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

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

Q19

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.