PROPERTIES OF TRIANGLE

About "Properties of triangle"

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.

2. 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 : The longest side of in the right triangle which is opposite to right angle (90°)

Properties of triangle - Practice 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.

Let us look at the next problem on "Properties of triangle"

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.

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

Let us look at the next problem on "Properties of triangle"

Problem 3 :

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

So, we have

x² = 8² + 6²

x² = 64 + 36

x² = 100

x = 10

Hence, the length of the hypotenuse is 10 units.

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Problem 4 :

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

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

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Problem 5 :

The sides of an equilateral triangle are shortened by 12 units, 13 units and 14 units respectively and a right angle triangle is formed. Find the length of each side of the equilateral triangle.

Solution :

Let "x" be the length of each side of the equilateral triangle.

Then, the sides of the right angle triangle are (x-12), (x-13) and (x-14)

In the above three sides, the side represented by (x -12) is hypotenuse (because that is the longest side).

Using Pythagorean theorem, (x-12)² = (x-13)² + (x-14)2

x² - 24x + 144 = x² - 26x + 169 + x² - 28x + 196

x² - 30x + 221 = 0

(x - 13)(x - 17) = 0

x = 13 or x = 17.

x = 13 can not be accepted. Because, if x = 13, one of the sides of the right angle triangle would be negative.

Hence, the side of the equilateral triangle is 17 units.

Let us look at the next problem on "Properties of triangle"

Problem 6 :

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°

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

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Problem 7 :

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°

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

Let us look at the next problem on "Properties of triangle"

Problem 8 :

In a triangle, If the second angle is 5° greater than the first angle and the third angle is 5° greater than second angle, find the three angles of the triangle.

Solution :

Let "x" be the first angle.

The second angle = x + 5

The third angle = x + 5 + 5 = x + 10

We know that,

the sum of the three angles of a triangle = 180°

x + (x+5) + (x+10) = 180°

3x + 15 = 180

3x = 165

x = 55

The first angle = 55°

The second angle = 55 + 5 = 60°

The third angle = 60 + 5 = 65°

Hence, the three angles of a triangle are 55°, 60° and 65°.

Let us look at the next problem on "Properties of triangle"

Problem 9 :

If the angles of a triangle are in the ratio 2 : 7 : 11, then find the angles.

Solution :

From the ratio 2 : 7 : 11,

the three angles are 2x, 7x, 11x

In any triangle, sum of the angles = 180°

So, 2x + 7x + 11x = 180°

20x = 180 -------> x = 9

Then, the first angle = 2x = 2(9) = 18°

The second angle = 7x = 7(9) = 63°

The third angle = 11x = 11(9) 99°

Hence the angles of the triangle are (18°, 63°, 99°)

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Problem 10 :

In a triangle, If the second angle is 10% more than the first angle and the third angle is 20% less than the first angle, then find the three angles of the triangle.

Solution :

Let "x" be the first angle.

The second angle = 120 % of x = 1.2x

The third angle = 80% of x = 0.8x

We know that,

the sum of the three angles of a triangle = 180°

x + 1.2x + 0.8x = 180°

3x = 180°

x = 60°

The first angle = 60°

The second angle = 1.2(60) = 72°

The third angle = 0.8(60) = 48°

Hence, the three angles of a triangle are 60°, 72° and 48°.

Let us look at the next problem on "Properties of triangle"

Problem 11 :

If 3 consecutive positive integers be the angles of a triangle, then find the three angles of the triangle.

Solution :

Let "x" be the first angle.

The second angle = x + 1

The third angle = x + 1 + 1 = x + 2

We know that,

the sum of the three angles of a triangle = 180°

x + x + 1 + x + 2 = 180°

3x + 3 = 180°

3x = 177°

x = 59°

The first angle = 59°

The second angle = 59 + 1 = 60°

The third angle = 60 + 1 = 61°

Hence, the three angles of a triangle are 59°, 60° and 61°.

Let us look at the next problem on "Properties of triangle"

Problem 12 :

In a triangle, if the second angle is 2 times the first angle and the third angle is 3 times the first angle, find the angles of the triangle.

Solution :

Let "x" be the first angle.

The second angle = 2x

The third angle = 3x

We know that,

the sum of the three angles of a triangle = 180°

x + 2x + 3x = 180°

6x = 180°

x = 30°

The first angle = 30°

The second angle = 2(30°) = 60°

The third angle = 3(30°) = 90°

Hence, the three angles of a triangle are 30°, 60° and 90°.

Let us look at the next problem on "Properties of triangle"

Problem 13 :

In a right triangle, apart from the right angle, the other two angles are x+1 and 2x+5. find the angles of the triangle.

Solution :

We know that,

the sum of the three angles of a triangle = 180°

90 + (x + 1) + (2x + 5) = 180°

3x + 6 = 90°

3x = 84°

x = 28°

So, x + 1 = 28 + 1 = 29°

2x + 5 = 2(28) + 5 = 56 + 5 = 61°

Hence, the three angles of a triangle are 90°, 29° and 61°.

Let us look at the next problem on "Properties of triangle"

Problem 14 :

In a triangle, if the second angle is 3 times the sum of the first angle and 3 and the third angle is the sum of 2 times the first angle and 3, find the three angles of the triangle.

Solution :

Let "x" be the first angle.

The second angle = 3(x+3)

The third angle = 2x + 3

We know that,

the sum of the three angles of a triangle = 180°

x + 3(x+3) + 2x + 3 = 180°

x + 3x + 9 + 2x + 3 = 180°

6x + 12 = 180°

6x = 168°

x = 28°

The first angle = 28°