What is the possible lengths for the third side of a triangle with sides 5 and 12?

Triangle is a closed figure which is formed by three line segments. It consists of three angles and three vertices. The angles of triangles can be the same or different depending on the type of triangle. There are different types of triangles based on line and angles properties.

Properties of a Triangle:

1. Each triangle has 3 sides and 3 angles.

2. Sum of all the angles of triangles is 180°

3. Perimeter of a triangle is the sum of all three sides of the triangle.

4. A triangle has 3 vertices.

Types of Triangles based on line Properties

Scalene Triangle: Scalene Triangle is a type of triangle in which all the sides are of different lengths. All the angles of a scalene triangle are different from one another.

Isosceles Triangle: Isosceles Triangle is another type of triangle in which two sides are equal and the third side is unequal. In this triangle, the two angles are also equal and the third angle is different.

Right-angled Triangle: A right-angled triangle is one that follows the Pythagoras Theorem and one angle of such triangles is 90 degrees which is formed by the base and perpendicular. The hypotenuse is the longest side in such triangles.

Equilateral Triangle: An equilateral triangle is a triangle in which all the three sides are of equal size and all the angles of such triangles are also equal.

Finding Third Side of a Triangle given Two Sides

Lets assume that the triangle is Right Angled Triangle because to find a third side provided two sides are given is only possible in a right angled triangle.

We know that the right-angled triangle follows Pythagoras Theorem

According to Pythagoras Theorem, the sum of squares of two sides is equal to the square of the third side. 

(Perpendicular)2 + (Base)2 = (Hypotenuse)2 

Using the above equation third side can be calculated if two sides are known.

Example: Suppose two sides are given one of 3 cm and the other of 4 cm then find the third side.

Lets take perpendicular P = 3 cm and Base B = 4 cm.

using Pythagoras theorem 

P2 + B2 = H2

(3)2 + (4)2 = H2

9 + 16 = H2

25 = H2

H = 5

Sample Questions

Question 1: Find the measure of base if perpendicular and hypotenuse is given, perpendicular = 12 cm and hypotenuse = 13 cm.

Solution: 

Perpendicular = 12 cm

Hypotenuse = 13 cm

Using Pythagoras Theorem 

P2 + B2 = H2

B2 = H2 – P2

B2 = 132 – 122

B2 = 169 – 144

B2 = 25

B = 5

Question 2: Perimeter of the equilateral triangle is 63 cm find the side of the triangle.

Solution: 

Perimeter of an equilateral triangle =  3×side

3×side = 64

side = 63/3

side = 21 cm

Question 3: Find the measure of the third side of a right-angled triangle if the two sides are 6 cm and 8 cm.

Solution: 

Perpendicular = 6 cm

Base = 8 cm

Using Pythagoras Theorem

H2 = P2 + B2 

H2 = P2 + B2

H2 = 62 + 82 

H2 = 36 + 64

H2 = 100

H = 10 cm

Question 4: Find whether the given triangle is a right-angled triangle or not, sides are 48, 55, 73?

Solution: 

A right-angled triangle follows the Pythagorean theorem so we need to check it .

Sum of squares of two small sides should be equal to the square of the longest side

so 482 + 552 must be equal to 732

2304 + 3025 = 5329 which is equal to 732 = 5329

Hence the given triangle is a right-angled triangle because it is satisfying the Pythagorean theorem.

Question 5: Find the hypotenuse of a right angled triangle whose base is 8 cm and whose height is 15 cm?

Solution: 

Using Pythagorean theorem, a2 + b2 = c2

So 82 + 152 = c2  

hence c = √(64 + 225)

          c = √289

          c = 17 cm

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