KITE QUESTION

Question

Kala and Ali are making a kite. To make the frame, they place a 40-inch stick horizontally across a 60-inch stick so that both sides of the horizontal stick are equal in length. Then they tie the two sticks together with the string to form right angles. The longer part of the 60-inch stick measures 50 inches. Kala says that 130 inches of string will be enough to stretch all around the kite frame. Ali says that they will need at least 153 inches of string. Who is correct? Explain.

Answer 1

Ali is correct. To determine the perimeter of the kite, draw a sketch. The horizontal stick is divided equally into two 20-inch lengths. The Vertical stick is divided 50 inches and 10 inches. Each side of the kite is the hypotenuse of a right triangle. There are two triangles that have legs of 10 inches and 20 inches. The other two triangles have legs of 50 and 20 inches.You must determine the length of each hypotenuse. Their sum is the perimeter of the kite. It represents how much string will be enough to stretch all of the way around the kite.

 

When I did this, I calculated 152.425 which is actually less than 153, but close enough. 

 

The 130 inches of string would be sufficient if the 60-inch stick was divided 30 and 30, instead of 50 and 10.

Answer 2

The angles created by the sticks make a series of right triangles.

2 triangles on top are equal

2 triangles on bottom are equal

 

The top triangles have one side of 20 inches and one side of 10 inches and one "unknown" side.

The bottom triangles have on side of 20 inches and one side of 50 inches and one "unknown" side.

 

Based on the question, I'm assuming you have been taught that any one side of a right triangle can be found if you know the other 2 sides using the Pythagorean Theorem.

 

That is:

 

a² + b² = c² ; where a and b are the 2 sides near the right angle (the ones we know) and c is the hypotenuse (the one we don't know).

 

Plug in the values of the a and b for one of the top triangles:

(20)² + (10)² = c²

400 + 100 = c²

500 = c²

Take the square root of both sides to find out what c is:

√500 = c

Factor the 500 into 100 * 5:

√(100*5) = c

√100*√5 = c ; by distribution rule

10√5 = c ; this is one side of the top; multiply by 2 to get both sides:

2*(10√5) == 20√5 that is both sides of the top.

 

Plug in the values for one bottom triangle:

(20)² + (50)² = c²

400 + 2500 = c²

2900 = c²

Again taking the square root of both sides:

√2900 = c

Factoring 2900 into 100 * 29:

√(100*29) = c

√100 * √29 = c ; by distribution rule

10√29 = c ; one side of bottom triangles

multiply by 2:

2*(10√29) == 20√29 ; both sides of the bottom

 

Now add the two together and you have the total perimeter of the kite:

20√5 + 20√29

 

Using a calculator to find the approximate amount of string to use:

20*(2.24) + 20*(5.39)

44.72 + 107.70 = 152.42 which is much closer to the approximate of 153 inches that Ali states.

 

Ali is correct!

 

 

Answer 3

The frame of the kite consists of 4 right triangles, two of them

(A and B) have legs of lengths 20" and 10", while the other two

(C and D) have legs of length 20" and 50".  The length of the string is the sum of the lengths of the hypotenuses of the 4 triangles.  

 

Hypotenuse of A has length sqrt(20² + 10²) = sqrt(500)

Hypotenuse of C has length sqrt(20² + 50²) = sqrt(2900)

 

Length of string = 2sqrt(500) + 2sqrt(2900) = 152.4"

 

Since some extra string is needed to tie the string to the sticks, Ali has the better estimate.