Question

The lengths of two sides of a triangle are $3\text{ cm}$and $4\text{ cm}$. Which of the following can be the length of the third side to form a triangle?
A. $0.5\text{ cm}$
B. $5\text{ cm}$
C. $\text{8 cm}$
D. $10\text{ cm}$

Answer 1

Hint: First we will describe what a triangle is. And after that we must describe the types of triangles based on the length of the sides and the type of angles and then we will write down the properties of the triangle and finally we will apply the property of a triangle that the sum of two sides is greater than the third side.

Complete step by step answer:
Let’s first understand what a triangle is. So, in geometry, a triangle is a closed, two-dimensional shape with three straight sides and three vertices. A triangle is also a polygon. Depending upon the sides and angles of a triangle, we have different types of triangles. Let’s discuss the types of triangles:
Based on the length of sides:
Scalene Triangle: All the sides and angles are unequal.

Isosceles Triangle: It has two equal sides. Also, the angles opposite these equal sides are equal.

Equilateral Triangle: All the sides are equal and all the three angles equal to 60.

Based on the type of angles:
Acute Angled Triangle: A triangle having all its angles less than ${{90}^{\circ }}$

Right Angled Triangle: A triangle having one of the three angles exactly ${{90}^{\circ }}$.

Obtuse Angled Triangle: A triangle having one of the three angles more than ${{90}^{\circ }}$

The properties of the triangle are:
A. The sum of all the angles of a triangle (of all types) is equal to ${{90}^{\circ }}$
B. The sum of the length of the two sides of a triangle is greater than the length of the third side. In the same way, the difference between the two sides of a triangle is less than the length of the third side.
C. The side opposite the greater angle is the longest side of all the three sides of a triangle.
D. The exterior angle of a triangle is always equal to the sum of the interior opposite angles. This property of a triangle is called an exterior angle property.
E. Two triangles are said to be similar if their corresponding angles of both triangles are congruent and lengths of their sides are proportional.
Now, we see that according to the second property of the triangle: The sum of the length of the two sides of a triangle is greater than the length of the third side and in the same way, the difference between the two sides of a triangle is less than the length of the third side.
So we are given one side of the triangle let’s say $AB=3\text{ cm}$ and $BC=4\text{ cm}$ , now we have to find out the length of $CA$ , as we can conclude from the property of a triangle : $AB+BC>CA\Rightarrow 3+4>CA\Rightarrow 7>CA$
So, $CA$ should be less than $7$ . Now from the options we can see that only $0.5\text{ and 5}$ are less than $7$ , now we see that with $0.5$ we can’t make the triangle, therefore $CA=5\text{ cm}$ .

Hence, the correct option is B.

Note:
Remember to always draw the diagrams after describing the types of triangles and also after finally finding the sides of the asked question. Always cover all the options and give reasons which options are rejected and why, for a better understanding of the examiner. Even if the student tries to draw a triangle with 0.5 cm as the third side, it would be impossible. The maximum or closest ones try to get, it will no longer represent a triangle. Consider the figure below,