Instead of the lengths being 40 cm, 60 cm, and 80 cm, suppose the sidelengths had been 4 cm, 6 cm, 8 cm (this triangle can fit on our page).

Question

Homi · Accepted Answer

We will explore the properties of a triangle with these specific side lengths.

Step 1 — Check if a triangle can be formed

A triangle must follow a special rule. The sum of any two sides must be greater than the third side. This rule is called the Triangle Inequality. Let us use $a = \mathbf{4 \text{ cm}}$ , $b = \mathbf{6 \text{ cm}}$ , $c = \mathbf{8 \text{ cm}}$ . We check all three conditions.

First condition: $a + b > c$ We add the first two sides. $4 + 6$ $= 10$ We compare this sum with the third side. $10 > 8$ This condition is true.

Second condition: $a + c > b$ We add the first and third sides. $4 + 8$ $= 12$ We compare this sum with the second side. $12 > 6$ This condition is true.

Third condition: $b + c > a$ We add the second and third sides. $6 + 8$ $= 14$ We compare this sum with the first side. $14 > 4$ This condition is true.

Since all three conditions are true, a triangle can be formed.

$\boxed{\text{A triangle can be formed.}}$

Diagram 1

Step 2 — Classify the triangle by its sides

We look at the lengths of the sides. If all three sides are different, it is a scalene triangle. If two sides are equal, it is an isosceles triangle. If all three sides are equal, it is an equilateral triangle. Our side lengths are $\mathbf{4 \text{ cm}}$ , $\mathbf{6 \text{ cm}}$ , and $\mathbf{8 \text{ cm}}$ . All three side lengths are different.

$\boxed{\text{The triangle is scalene.}}$

Step 3 — Classify the triangle by its angles

We can classify a triangle by its largest angle. Let $a$ , $b$ , $c$ be the side lengths. Let $c$ be the longest side. If $a^2 + b^2 = c^2$ , it is a right-angled triangle. If $a^2 + b^2 > c^2$ , it is an acute-angled triangle. If $a^2 + b^2 < c^2$ , it is an obtuse-angled triangle. The longest side is $c = \mathbf{8 \text{ cm}}$ . Let us use $a = \mathbf{4 \text{ cm}}$ and $b = \mathbf{6 \text{ cm}}$ .

Let us calculate $a^2 + b^2$ . $4^2 + 6^2$ $= 16 + 36$ $= 52$

Now let us calculate $c^2$ . $8^2$ $= 64$

We compare $52$ and $64$ . $52 < 64$ Since $a^2 + b^2 < c^2$ , the triangle is obtuse-angled.

$\boxed{\text{The triangle is obtuse-angled.}}$

Step 4 — Congruence with another triangle

Congruent triangles are exact copies of each other. They have the same shape and size. The SSS (Side-Side-Side) rule is important. It says: if one triangle has sides equal to another triangle's sides. Then these two triangles are congruent. Let us consider a second triangle. Its side lengths are $\mathbf{4 \text{ cm}}$ , $\mathbf{6 \text{ cm}}$ , and $\mathbf{8 \text{ cm}}$ . Its sides are equal to the sides of our first triangle. So, by the SSS rule, the two triangles are congruent.

$\boxed{\text{Any two triangles with these side lengths are congruent.}}$

Answer

(i) The triangle is scalene. (ii) The triangle is obtuse-angled. (iii) Any two triangles with these side lengths are congruent by the SSS rule.

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