**Finding measurements of a triangle :**

In this section, we are going to study, how to find the area, perimeter of the triangle with known and unknown measures. And also, we can find the unknown measures, when we know the area and perimeter of the triangle.

For example, to find the area of a triangle, we have to know the base and height. In some problems, height will be given and base will not be given. We have to use the other information to find the base of the triangle.

**Example 1 :**

Find the area and perimeter of the triangle shown below.

**Solution : **

**Because we want to find the area of the triangle, we have to know its base and height. **

To know the base and height of the triangle, let us rotate the given triangle as shown below.

Base = 21 units

Height = 8 units

Area of the triangle = (1/2)⋅base⋅height

Area of the triangle = (1/2)⋅21⋅8

Area of the triangle = 84 square units

Perimeter = Sum of the lengths of all the three sides

Perimeter = 17 + 10 + 21

Perimeter = 48 units

**Example 2 :**

Find the area and perimeter of the triangle given below.

**Solution : **

**Because we want to find the area of the triangle, we have to know its base and height. **

To know the base and height of the triangle, let us rotate the given triangle as shown below.

Use Pythagorean theorem to find the height of the triangle.

h² + 5² = (5√2)²

h² + 25 = 25⋅2

h² + 25 = 50

Subtract 25 from both sides

h² = 25

h² = 5²

h = 5

So, height = 5 units.

Area of the triangle = (1/2)⋅base⋅height

Area of the triangle = (1/2)⋅5⋅5

Area of the triangle = 12.5 square units

Perimeter = Sum of the lengths of all the three sides

Perimeter = 5 + 5 + 5√2

Perimeter = (10 + 5√2) units

**Example 3 :**

Area of the triangle shown below is 126 square units. Find the perimeter and height of the triangle.

For our convenience, let us rotate the given triangle as shown below.

From the triangle above,

Base = 21 units

Given :

Area of the triangle = 126 square units

(1/2)⋅base⋅height = 126

Substitute base = 21.

(1/2)⋅21⋅height = 126

Multiply both sides by 2.

21⋅height = 126⋅2

21⋅height = 252

Divide both sides by 21.

(21⋅height) / 21 = 252 / 21

height = 12

So, height of the triangle is 12 units.

Perimeter = Sum of the lengths of all the three sides

Perimeter = 13 + 20 + 21

Perimeter = 54 units

**Example 4 :**

Find the area and perimeter of the triangle defined by D(1, 3), E(8, 3) and F(4, 7).

**Solution : **

Plot the points in the coordinate plane. Draw the height from F to the side DE. Label the point where the height meets DE as G. Point G has coordinates (4, 3).

Base = 8 - 1 = 7 units

Height = 7 - 3 = 4 units

Area of the triangle = (1/2)⋅base⋅height

Area of the triangle = (1/2)⋅7⋅4

Area of the triangle = 14 square units

To find the perimeter, we have to know the lengths of all the three sides.

Length of the side DE = 8 - 1 = 7 units.

Use distance formula to find the lengths of EF and FD.

**Distance formula : **

**Finding the length of EF :**

(x₁, y₁) = E(8, 3)

(x₂, y₂) = F(4, 7)

Then, we have

EF = √[(4 - 8)² + (7 - 3)²]

EF = √[(-4)² + (4)²]

EF = √[16 + 16]

EF = √32

EF = 4√2 units

**Finding the length of FD :**

(x₁, y₁) = F(4, 7)

(x₂, y₂) = D(1, 3)

Then, we have

FD = √[(1 - 4)² + (3 - 7)²]

FD = √[(-3)² + (-4)²]

FD = √[9 + 16]

FD = √25

FD = 5 units

**Perimeter of the triangle :**

Perimeter = Sum of the lengths of all the three sides

Perimeter = FD + DE + EF

Perimeter = 5 + 7 + 4√2

Perimeter = (12 + 4√2) units

After having gone through the stuff given above, we hope that the students would have understood "Finding measurements of a triangle".

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