PROPERTIES OF TRIANGLE

The following are the two important properties of triangle.

1. The sum of the lengths of any two sides of a triangle is greater than the third side.

2. The sum of all the three angles of a triangle is 180°.

Some Other Properties of Triangle

1. In an equilateral triangle, all the three sides and three angles will be equal and each angle will measure 60°.

2. In an isosceles triangle, the lengths of two of the sides will be equal. And the corresponding angles of the equal sides will be equal.

3. In a right triangle, square of the hypotenuse is equal to the sum of the squares of other two sides. This is known as Pythagorean theorem.

Hypotenuse :

Hypotenuse is the longest side in any right triangle which is opposite to  right angle (90°).

Solved Problems

Problem 1 :

Is it possible to have a triangle whose sides are 5 cm, 6 cm and 4 cm?

Solution :

According to the properties of triangle explained above, if the sum of the lengths of any two sides is greater than the third side, then the given sides will form a triangle.

Let us apply this property for the given sides.

5 cm + 6 cm > 4 cm

6 cm + 4 cm > 5 cm

5 cm + 4 cm > 6 cm

Since the given sides meet the condition said in the property, It is possible to have a triangle whose sides are 5 cm, 6 cm and 4 cm.

Problem 2 :

Is it possible to have a triangle whose sides are 7 cm, 2 cm and 4 cm?

Solution :

According to the properties of triangle explained above, if the sum of the lengths of any two sides is greater than the third side, then the given sides will form a triangle.

Let us apply this property for the given sides.

2 cm  + 4 cm < 7 cm

From the above point, it is clear the sum of the lengths of the two sides 2 cm and 4 cm is  less than the third side 7 cm.

The given sides do not meet the condition said in the property.

So, it is not possible to have a triangle whose sides are 7 cm, 2 cm and 4 cm.

Problem 3 :

Find the range of possible measures of x in the following given sides of a triangle :

10, 7, x

Solution :

Lengths of the given two sides are 6 and 7.

Difference of the lengths = 7 - 6 = 1

Sum of the lengths = 7 + 6 = 13

Hence, the range of possible measures for the third side x is

1 < x < 13

Problem 4 :

Can 30°, 60° and 90° be the angles of a triangle?

Solution :

Let us add all the three given angles and check whether the sum is equal to 180°.

 30° + 60° + 90° = 180°

Because the sum of the angles is equal 180°, the given three angles can be the angles of a triangle.

Problem 5 :

Can 35°, 55° and 95° be the angles of a triangle?

Solution :

Let us add all the three given angles and check whether the sum is equal to 180°.

 35° + 55° + 95° = 185°

Because the sum of the angles is not equal 180°, the given three angles can not be the angles of a triangle.

Problem 6 :

Find the length of the hypotenuse of the right triangle where the lengths of the other two sides are 8 units and 6 units.

Solution :

From the given information we can draw the triangle as given below.

In the above triangle, we have to find the value of x.

According to Pythagorean theorem, square of the hypotenuse is equal to the sum of the squares of other two sides.

x² = 8² + 6²

x² = 64 + 36

x² = 100

x = 10

The length of the hypotenuse is 10 units.

Problem 7 :

The hypotenuse of a right angled triangle is 20 cm. The difference between its other two sides is 4 cm. Find the length of the sides.

Solution :

Let x and x + 4 be the lengths of other two sides.

Using Pythagorean theorem,

(x + 4)² + x² = 202

x² + 8x + 16 + x² - 400 = 0

2x² + 8x - 384 = 0

x² + 4x - 192 = 0

(x + 16)(x - 12) = 0

x = -16 or x = 12

x = -16 can not be accepted. Because length can not be negative.

If x = 12,

x + 4 = 12 + 4 = 16

The other two sides of the triangle are 12 cm and 16 cm.

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