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In a right triangle, the longer leg is two more than three times the s [#permalink]
gvij2017 wrote:
l = 3s + 2
Area of triangle: ls/2
400 = s*(3s+2)/2
3s^2 + 2s = 800
s=16 is answer to this equation.

Is there any convenient way to solve the equation \(3s^2 + 2s = 800\) ? I transformed it to \(s² + \frac{2b}{3}- \frac{800}{3}\) and used the pq-method. The problem is that I had to find \(\sqrt{2401}\), which equals 49. However, determining that is it 49 is not that simple.

Thats why I am asking whether there is an easier way. Thanks for any help!
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Re: In a right triangle, the longer leg is two more than three times the s [#permalink]
Masterscorp wrote:
gvij2017 wrote:
l = 3s + 2
Area of triangle: ls/2
400 = s*(3s+2)/2
3s^2 + 2s = 800
s=16 is answer to this equation.

Is there any convenient way to solve the equation \(3s^2 + 2s = 800\) ? I transformed it to \(s² + \frac{2b}{3}- \frac{800}{3}\) and used the pq-method. The problem is that I had to find \(\sqrt{2401}\), which equals 49. However, determining that is it 49 is not that simple.

Thats why I am asking whether there is an easier way. Thanks for any help!


The OA suggested to go by back solving with the answer choices. Started with (C) which eliminated (D) and (E) because (D) and (E) greater than (C). Then A and B was tested.
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In a right triangle, the longer leg is two more than three times the s [#permalink]
1
Kudos
Expert Reply
Masterscorp wrote:
gvij2017 wrote:
l = 3s + 2
Area of triangle: ls/2
400 = s*(3s+2)/2
3s^2 + 2s = 800
s=16 is answer to this equation.

Is there any convenient way to solve the equation \(3s^2 + 2s = 800\) ? I transformed it to \(s² + \frac{2b}{3}- \frac{800}{3}\) and used the pq-method. The problem is that I had to find \(\sqrt{2401}\), which equals 49. However, determining that is it 49 is not that simple.

Thats why I am asking whether there is an easier way. Thanks for any help!

Masterscorp , the \(pq\) method is not easy here. If you can use the pq method, the two suggestions below will be easy.

To find the square root of any difficult perfect square, you can

1) think about easy perfect squares that are "close" to 2401:
\(2500 = 50^2\)
\(1600 = 40^2\)
The square root of 2401 is between 40 and 50

2500 and its square root, 50, are too great, but not by much. Try 49. \(49^2=2401\)

OR: the units digit of \(\sqrt{2401}\) is either 1 or 9
Try 49 (i.e., instead of 41, because \(41^2\) will be too close to 1600)

2) Find prime factors of 2401

2401 is not divisible by 2, 3, or 5. Try 7:
\(\frac{2401}{7}=343\)
343 is not divisible by 2, 3, or 5. Try 7:
\(\frac{343}{7}=49=(7*7)\)
We have 2401 = (7*7*7*7) = (49*49), that is
\(\sqrt{2401}=\sqrt{49*49}=49\)
Done

Wanting to understand is smart.

On other occasions, JMO, it's better to use a mix of understanding and available shortcuts - in this case, some equation (not necessarily a quadratic, which I avoided) + answer choices.

Hope that helps. :-)
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In a right triangle, the longer leg is two more than three times the s [#permalink]
gvij2017 wrote:
l = 3s + 2
Area of triangle: ls/2
400 = s*(3s+2)/2
3s^2 + 2s = 800
s=16 is answer to this equation.


How do we know that the second leg is actually the gratest?

I mean, \(\frac{bh}{2}\) = 400

But it's written that the greatest leg is trice times the smalles one plus two.

Why is it correct to put it like this?

400 = s*(3s+2)/2

Shouldn´t the greatest be the hipotenuse?

Kind regards!
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Re: In a right triangle, the longer leg is two more than three times the s [#permalink]
1
Kudos
Solving this quadratically wasn't really feasible for me, so I solved by plugging in the answer choices.

Started with B, found the answer was too high (620) so reduced it by selecting a smaller x (A)
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Re: In a right triangle, the longer leg is two more than three times the s [#permalink]
jfranciscocuencag wrote:
gvij2017 wrote:
l = 3s + 2
Area of triangle: ls/2
400 = s*(3s+2)/2
3s^2 + 2s = 800
s=16 is answer to this equation.


How do we know that the second leg is actually the gratest?

I mean, \(\frac{bh}{2}\) = 400

But it's written that the greatest leg is trice times the smalles one plus two.

Why is it correct to put it like this?

400 = s*(3s+2)/2

Shouldn´t the greatest be the hipotenuse?

Kind regards!


Leg of a Right Triangle. Either of the sides in a right triangle opposite an acute angle. The legs are the two shorter sides of the triangle, hypotenuse is not consider a leg.
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Re: In a right triangle, the longer leg is two more than three times the s [#permalink]
generis wrote:
anupam87 wrote:
In a right triangle, the longer leg is two more than three times the shorter leg, and the area of the triangle is 400. What is the length of the shorter leg?

A 16
B 20
C 25
D 32
E 40

Using answer choices
Short side = S
Long side = L = (3S + 2)
Area: \(\frac{S * L}{2}=400\)
AND: \(S*L=800\)


Shortcut: mental math. Multiply each answer by 3 and add 2. The result must be a factor of 800 (see area, above)

Eliminate B (62), C (77), D (98), and E (122)
Option A = 50, which is a factor of 800

Answer A

Standard approach
Alternatively, start with C.

Answer C) \(25 = S\)
Long side: \(L=(3S + 2) =((25*3)+2))=77\)
Approximate area,
\(A=\frac{S*L}{2}= \frac{25*76}{2}=\frac{50*38}{2}=\frac{100*19}{2}=\frac{1900}{2}\approx{950}\)


Actual area is \(400.\) - \(950\) is much too great. Eliminate D and E. Both are greater than C. Try A or B? 950 is 2+ times 400
Both sides need to be a lot shorter

Try A) \(S = 16\)
\(L=((16*3)+2)=(48+2)=50\)

Area = \(\frac{S*L}{2}=\frac{16*50}{2}=\frac{800}{2}=400\)

That's correct

Answer A


Damm, why did I not see the first approach, brilliant and easy way to solve this question. Kudos!
As many others were, I got stuck at the quadratic equation - I knew it was not the intent of the person who wrote this question to calculate such a crappy quadratic equation with out a calculator.
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Re: In a right triangle, the longer leg is two more than three times the s [#permalink]
Expert Reply
Hi All,

We're told that in a right triangle, the longer leg is two more than three times the shorter leg, and the area of the triangle is 400. We're asked for the length of the SHORTER leg. This question can be solved in a couple of different ways - and it can be solved rather easily by TESTing THE ANSWERS.

To start, we know that the area of the triangle is 400, so we can set up the formula for Area:
A = (1/2)(Base)(Height)
400 = (1/2)(Base)(Height)
800 = (Base)(Height)

We're asked for the SHORTER side, so we should start with one of the smaller answers. Let's TEST Answer B first....

Answer B: 20
IF... the shorter side = 20

the longer leg is two more than three times the shorter leg....
longer side = 2 + (3)(20) = 62

With side lengths of 20 and 62, the product would be (20)(62) = 1240. This is clearly too big (it's supposed to be 800). Thus, we need the sides the be SMALLER. There's only one answer that fits....

Final Answer:

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Re: In a right triangle, the longer leg is two more than three times the s [#permalink]
generis wrote:
anupam87 wrote:
In a right triangle, the longer leg is two more than three times the shorter leg, and the area of the triangle is 400. What is the length of the shorter leg?

A 16
B 20
C 25
D 32
E 40

Using answer choices
Short side = S
Long side = L = (3S + 2)
Area: \(\frac{S * L}{2}=400\)
AND: \(S*L=800\)


Shortcut: mental math. Multiply each answer by 3 and add 2. The result must be a factor of 800 (see area, above)

Eliminate B (62), C (77), D (98), and E (122)
Option A = 50, which is a factor of 800

Answer A

Standard approach
Alternatively, start with C.

Answer C) \(25 = S\)
Long side: \(L=(3S + 2) =((25*3)+2))=77\)
Approximate area,
\(A=\frac{S*L}{2}= \frac{25*76}{2}=\frac{50*38}{2}=\frac{100*19}{2}=\frac{1900}{2}\approx{950}\)


Actual area is \(400.\) - \(950\) is much too great. Eliminate D and E. Both are greater than C. Try A or B? 950 is 2+ times 400
Both sides need to be a lot shorter

Try A) \(S = 16\)
\(L=((16*3)+2)=(48+2)=50\)

Area = \(\frac{S*L}{2}=\frac{16*50}{2}=\frac{800}{2}=400\)

That's correct

Answer A




Hi, I didnt quite understand why you were doing shorter side* longer side/2? when is this formula used?
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Re: In a right triangle, the longer leg is two more than three times the s [#permalink]
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Hi Jaya6,

The formula for Area of a Triangle is:

A = (1/2)(Base)(Height)

Here, since we're dealing with a Right Triangle, we can make the two 'legs' of the triangle the Base and the Height. Multiplying a product by 1/2 is the same as dividing that product by 2 - and if you're going to take a calculation-heavy approach (instead of a faster, Tactical one), then since the area is 400, you would eventually end up multiplying the two legs of the triangle and then dividing by 2 (since that product equals 400) and then going through the additional Algebraic steps to solve for the shorter side.

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In a right triangle, the longer leg is two more than three times the s [#permalink]
dcummins wrote:
Solving this quadratically wasn't really feasible for me, so I solved by plugging in the answer choices.

Started with B, found the answer was too high (620) so reduced it by selecting a smaller x (A)

I started with the answer choices as well.
First, wrote down the formula for area.
\(\frac{1}{2}leg_1*leg_2=400 \implies leg_1*leg_2=800\)
Now, started testing answer choices - started with C & found a match in A.


OptionL1L2L1*L2
C2577\(\ne800\)
B2062\(\ne800\)
A1650800
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Re: In a right triangle, the longer leg is two more than three times the s [#permalink]
Short leg = 16, then larger leg = 3(16) + 2 = 50

Then A = (0.5)bh = (0.5)(16)(50) = (8)(50) = 400 Answer
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Re: In a right triangle, the longer leg is two more than three times the s [#permalink]
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