Exercise 90 Page 514

For the given trapezoid, we want to find both the area and perimeter.

Area

The area of a trapezoid is calculated by multiplying its height with the sum of the parallel sides, divided by 2. A=1/2h(b_1+b_2) Let's identify these dimensions in the trapezoid and calculate the area.

Perimeter

The perimeter is the sum of the trapezoid's sides. We need to find the length of the non-parallel sides, which are unknown. Note that the triangle on the left-hand side of the trapezoid is a right triangle where one of the non-right angles is 45^(∘). Since we know two angles in this triangle we can find the last one, which we will label θ, by using the Triangle Angle Sum Theorem. θ+45^(∘)+90^(∘)=180^(∘) Let's solve for θ in this equation.

θ+45^(∘)+90^(∘)=180^(∘)

AddTerms

Add terms

θ+135^(∘)=180^(∘)

SubEqn

LHS-135^(∘)=RHS-135^(∘)

θ=45^(∘)

Since two of the triangles angles have equal measures, we know by the Base Angles Theorem that this is an isosceles triangle. With this we can add some information to the diagram.

Now we have enough information to calculate one of the non-parallel sides with the Pythagorean Theorem.

a^2+b^2=c^2

SubstituteII

a= 4, b= 4

4^2+ 4^2=c^2

▼

Solve for c

CalcPow

Calculate power

16+16=c^2

AddTerms

Add terms

32=c^2

RearrangeEqn

Rearrange equation

c^2=32

SqrtEqn

sqrt(LHS)=sqrt(RHS)

c=± sqrt(32)

c > 0

c=sqrt(32)

To find the length of the last unknown side,we note that it also can be viewed as the hypotenuse of a right triangle with legs of 3 and 4.

Again, let's use the Pythagorean Theorem to calculate the last unknown side.

a^2+b^2=c^2

SubstituteII

a= 3, b= 4

3^2+ 4^2=c^2

▼

Solve for c

CalcPow

Calculate power

9+16=c^2

AddTerms

Add terms

25=c^2

RearrangeEqn

Rearrange equation

c^2=25

SqrtEqn

sqrt(LHS)=sqrt(RHS)

c=± 5

c > 0

c=5

Now we have everything we need to calculate the perimeter.