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Intern  S
Joined: 29 Dec 2018
Posts: 31
The hypotenuse of a right triangle is 16 ft longer than the length of  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 61% (02:55) correct 39% (03:22) wrong based on 33 sessions

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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
Math Expert V
Joined: 02 Aug 2009
Posts: 8005
Re: The hypotenuse of a right triangle is 16 ft longer than the length of  [#permalink]

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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

(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 = $$\sqrt{24^2+10^2}=26$$

A
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Intern  B
Joined: 11 Oct 2018
Posts: 21
Location: Germany
The hypotenuse of a right triangle is 16 ft longer than the length of  [#permalink]

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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 $$a=10, b=24, c=26$$.

$$a+b>c...10+24=34>26$$
$$a+c>b...10+26=36>24$$
$$b+c>a...24+26=50>10$$ The hypotenuse of a right triangle is 16 ft longer than the length of   [#permalink] 11 Mar 2019, 06:46
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