Algebra 1 Question

Question

One base of a trapezoid is three times the length of the second base. The height of the trapezoid is 2in. smaller than the second base. If the area of the trapezoid is 30in^2, find the lengths of the bases and the height of the trapezoid.

Answer 1

Hi Alice,

Let's use the variable b₁ to represent the first base and the variable b₂ tor the second base. Then we can use the variable h for the height.

The formula for calculating the area of a trapezoid is:

A = (b₁ + b₂ ) h

2

Since the length of b₁ is 3 times that of b₂, we can write: b₁ = 3b₂

Also, from the info in the problem, we can represent the height as : h= b₂ - 2

Using these expressions for b₁ and h and knowing that the area is 30, we can substitute these expressions into the formula and solve for b₂:

30 = (3b₂ + b₂) (b₂ - 2 )

2

Now combine like terms and use inverse operations and factoring to find the value of b₂:

30 = 4b₂ (b₂ - 2)

2

30 • 2 = 4b₂ (b₂ - 2) • 2

2

60 = 4b₂² - 8 b₂

0 = 4b₂² - 8 b₂ - 60

0 = 4 (b₂² - 2 b₂ - 15)

0 = 4 • (b₂ - 5 )(b₂ + 3)

Since the base can not be a negative number:

(b₂ - 5 ) = 0

b₂ = 5 inches

b₁ = 3b₂ = 3 • 5 =15 inches

h = b₂ -2 = 5 - 2 = 3inches

To verify:

A = (b₁ + b₂) h

2

A = (15 + 5) 3 = 20 • 3 = 30 square inches

2 2

Answer 2

Hi!

First let’s identify the variables.

Let B= 1st base

let b = 2nd base

let h= height

Now we need to break down each statement and make it an algebraic statement using the variable we identified.

One base of a trapezoid is three times the length of the second base. So we could say B= 3b

The height of the trapezoid is 2in. smaller than the second base. So we could say h= b-2

Area =30 or A=30

Now we will use the formula for Area of a trapezoid:

A= (h(B+b) ) / 2

Then we will replace A, B and h into the formula.

30 = (b-2)(3b+b) /2

Multiplying both sides by 2 and combining like terms you now get:

60=(b-2)(4b)

Next distribute the right side ( multiplying 4b to b and -2)

60= 4b²-8b

Next divide both sides by 4

15= b²-2b

Next -15 to both sides to make left side 0

0= b² -2b -15

Now factor; ( find two numbers that multiply to make -15 and add to make -2 ans: -5 and 3

0=(b-5)(b+3)

Set each factor to 0 and solve for b

b-5=0 b+3=0

b=5 or b=-3 [b=-3 doesn’t work only positive sol do in geometry bc it’s length. ]

Now that you know b you can now find B and h

h=b-2 B= 3b

h= 5-2. B=3(5)

h=3 B= 15

h=3

To have an area of 30 that means b= 5 B = 15 and h =3

I hope this helps! Have a nice evening!

Mrs. S