Exercise 39 Page 627

The area of a rhombus is half the product of its diagonals. Recall that the diagonals of a rhombus are perpendicular. We also know that the diagonals bisect each other. That gives us that the two halves of the horizontal diagonal are congruent. By the definition of congruence, they have the same length.

Let's consider the right triangle with an acute angle of 30^(∘).

By using the tangent ratio we can calculate the length of the opposite side of this triangle. Let's do it!

tan(θ)=Opp./Adj.

SubstituteValues

Substitute values

tan(30^(∘))=a/8

MultEqn

LHS * 8=RHS* 8

tan(30^(∘))* (8)=a

RearrangeEqn

Rearrange equation

a=tan(30^(∘))* (8)

We can write the length of a in radical form by using the properties of special right triangles. a=tan(30^(∘))* (8) ⇕ a=sqrt(3)/3* (8)=8sqrt(3)/3 Now we know the length of the opposite side. Once more we will use that the diagonals bisect each other.

By using the Segment Addition Postulate we can now find the lengths of the diagonals. d_1:& 8+8= 16in. d_2:& 8sqrt(3)/3+8sqrt(3)/3=16sqrt(3)/3m Using the diagonal lengths, let's find the area of the rhombus.

A=1/2d_1 d_2

SubstituteII

d_1= 16, d_2= 16sqrt(3)/3

A=1/2( 16)(16sqrt(3)/3)

▼

Evaluate right-hand side

Multiply

Multiply

A=(8) 16sqrt(3)/3

MoveLeftFacToNum

a*b/c= a* b/c

A= (8)16sqrt(3)/3

Multiply

Multiply

A=128sqrt(3)/3

The area of the rhombus is 128sqrt(3)3in^2.