Question

A parallelogram has two 19-inch sides and two 23-inch sides. What is the range of possible lengths for the diagonals of this parallelogram?

Answer 1

Answer:

Its typically helpful to start with a drawing. Here is

A

B

C

D

as given by the problem.

Geogebra

Geogebra

We are looking for the length of the shorter diagonal, which is segment

B

D

. This segment forms a triangle with the two known sides. Since we know two sides and the angel connecting them, we can use the law of cosines to solve for the unknown segment.

http://mathworld.wolfram.com/LawofCosines.html

http://mathworld.wolfram.com/LawofCosines.html

The law of cosines tells us that

c

2

=

a

2

+

b

2

−

2

a

b

cos

(

C

)

for the triangle labeled above. If we choose our two known sides for

a

and

b

and our known angle for

C

, we can solve for the length of the diagonal,

c

.

c

2

=

14

2

+

8

2

−

2

(

14

)

(

8

)

cos

(

60

o

)

≈

12.17

Step-by-step explanation:

sorry about the last answer i thought i was on another question

Answer 2

Answer:

• Between 4 and 42 inches

Step-by-step explanation:

The two sides and diagonal make a triangle.

Let the diagonal is d, use triangle inequality:

• 19 + 23 > d ⇒ 42 > d ⇒ d < 42
• 19 + d > 23 ⇒ d > 23 - 19 ⇒ d > 4
• 23 + d > 19 ⇒ d > 0

Solution is:

• 4 < d < 42