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The hypotenuse of a right triangle is 16 ft longer than the length of

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MartinTao
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The hypotenuse of a right triangle is 16 ft longer than the length of [#permalink] New post   07 Mar 2019, 00:24
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The hypotenuse of a right triangle is 16 ft longer than the length of the shorter leg. If the area of this triangle is exactly 120 ft², what is the length of the hypotenuse in feet?

A. 26

B. 32

C. 40

D. 64

E. 80
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A
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chetan2u
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Re: The hypotenuse of a right triangle is 16 ft longer than the length of [#permalink] New post   07 Mar 2019, 01:25
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MartinTao wrote:
The hypotenuse of a right triangle is 16 ft longer than the length of the shorter leg. If the area of this triangle is exactly 120 ft², what is the length of the hypotenuse in feet?

A. 26

B. 32

C. 40

D. 64

E. 80



Two ways..
(1) I would recommend use of choices as a priority in such a question.
Let the hypotenuse be option A, that is 26, so one leg = 26-16=10....
So the other side = \(\sqrt{26^2-10^2}=24\)
Thus, the area = (24*10)/2=120... TRUE
so answer A

(2) Algebraic method
Let the legs be x and y, so hypotenuse = x+16 and area xy/2=120 or xy=240
by Pythagorean theorem \((x+16)^2=x^2+y^2.....x^2+32x+256=x62+y^2.......y^2=32x+256\) substitute x= 240/y from xy=240
Thus, \(y^2=32*\frac{240}{y}+256.......y^3-256y=32*240\)
We are not asked cubic functions, that is power of 3, but we can use \(y(y^2-256)=32*240.......(y-16)(y+16)y=32*240=8*24*40=(24-16)*24*(24+16)\)
so y = 24, x= 240/24=10 and hypotenuse = \(\sqrt{24^2+10^2}=26\)

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The hypotenuse of a right triangle is 16 ft longer than the length of [#permalink] New post   11 Mar 2019, 06:46
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MartinTao wrote:
The hypotenuse of a right triangle is 16 ft longer than the length of the shorter leg. If the area of this triangle is exactly 120 ft², what is the length of the hypotenuse in feet?

A. 26

B. 32

C. 40

D. 64

E. 80


Another way is working with triangle properties.
We know that the sum of two sides must be greater than the third side. Let's work with this:

short leg: a
long leg: b
hypotenuse: c=a+16

This means, the long leg has to be longer than 16 ft.

Now, we factorize 240 (which is a*b).
\(240=2^4*3*5\)
\(2^4\) is \(16\), but we need greater than \(16\), so lets try \(2^3*3=24=b\).
In this case \(a=10, b=24, c=26\).

\(a+b>c...10+24=34>26\)
\(a+c>b...10+26=36>24\)
\(b+c>a...24+26=50>10\)

Answer (A)
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Re: The hypotenuse of a right triangle is 16 ft longer than the length of [#permalink] New post   22 Oct 2020, 23:25
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MartinTao wrote:
The hypotenuse of a right triangle is 16 ft longer than the length of the shorter leg. If the area of this triangle is exactly 120 ft², what is the length of the hypotenuse in feet?

A. 26

B. 32

C. 40

D. 64

E. 80


Area = (1/2) *Leg1 * Leg2 = 120

Leg1 * Leg2 = 240
The legs can be 1, 240 or 2, 120... or 10, 24 or 12, 20 or 15, 16 etc.. Note that the difference between the two legs must be less than 16 because hypotenuse is the largest side.

The length of the hypotenuse can be 10+16 = 26 or 12+16 = 28 etc.
Since all sides are integers, we will get a pythagorean triplet so we do not need to calculate anything.
10, 24, 26 is a triplet (a multiple of 5, 12, 13)

Answer (A)
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Re: The hypotenuse of a right triangle is 16 ft longer than the length of [#permalink] New post   23 Oct 2020, 21:07
1st thing to look for is if there are any Common Pythagorean Triplets Present (especially since the Answer Choices are nice whole Numbers)


Shorter Leg = X

Longer Leg = Y

Hypotenuse = X + 16

X * Y = 240

-E- 80

X = 80 - 16 = 64

Y is NOT the Longer Leg so this Choice is NOT Possible


-D- 64

X = 64 - 16 = 48

Y = 240/X --------> Y = 240/48 = 5

Again, Y is NOT the Longer Leg. This Choice is NOT Possible


-C- 40

X = 40 - 16 = 24

Y = 240/X ------> 240/24 = 10


Y again is NOT the Longer Leg ---- this Choice is NOT Possible




Only Answers A and B are Possible:

-B- 32

X = 32 - 16 = 16

Y = 240/16 = 15


15 - 16 - 32


OR


-A- 26

X = 26 - 16 = 10

Y = 240/X = 240/10 = 24


10 - 24 - 26 --------> taking 2 Common from Each Side Length

2*(5) - 2*(12) - 2*(13)

we have a Multiple of a 5 - 12 - 13 Triangle.

Answer Choice A must Work

-A-

-A-
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Re: The hypotenuse of a right triangle is 16 ft longer than the length of [#permalink] New post   26 Oct 2020, 08:41
VeritasKarishma wrote:
MartinTao wrote:
The hypotenuse of a right triangle is 16 ft longer than the length of the shorter leg. If the area of this triangle is exactly 120 ft², what is the length of the hypotenuse in feet?

A. 26

B. 32

C. 40

D. 64

E. 80


Area = (1/2) *Leg1 * Leg2 = 120

Leg1 * Leg2 = 240
The legs can be 1, 240 or 2, 120... or 10, 24 or 12, 20 or 15, 16 etc.. Note that the difference between the two legs must be less than 16 because hypotenuse is the largest side.

The length of the hypotenuse can be 10+16 = 26 or 12+16 = 28 etc.
Since all sides are integers, we will get a pythagorean triplet so we do not need to calculate anything.
10, 24, 26 is a triplet (a multiple of 5, 12, 13)

Answer (A)


Why are you assuming the legs must be integers?
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Re: The hypotenuse of a right triangle is 16 ft longer than the length of [#permalink] New post   17 Oct 2023, 12:35
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