SOLUTION: the altitude to the hypotenuse of a right triangle divides the hypotenuse in the ratio 4:1. what is the ratio of legs of the triangle?

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Question 1206000: the altitude to the hypotenuse of a right triangle divides the hypotenuse in the ratio 4:1. what is the ratio of legs of the triangle?
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52756)   (Show Source): You can put this solution on YOUR website!
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the altitude to the hypotenuse of a right triangle divides the hypotenuse in the ratio 4:1.
what is the ratio of legs of the triangle?
~~~~~~~~~~~~~~~~~~~~~~~~~ 

Let x and y be the segments the altitude divides the hypotenuse.


Then the altitude drawn to the hypotenuse is  the mean geometric value of x and y,  h = .   


    +---------------------------------------------------------------------+
    |    This remarkable formula is from the basic course of Geometry.    |
    |                  It is your pre-requisite.                          |
    |   It is assumed you know it from your previous course of Geometry.  |
    +---------------------------------------------------------------------+


We are given that y = 4x;  so,  h =  =  = 2x.

Thus the ratio    is   = 2.


           +--------------------------------------------------+
           |      In other words, the ratio of the legs       |
           |    of the smallest right-angled triangle is 2.   |
           +--------------------------------------------------+


The smallest triangle of the division is SIMILAR to the original large triangle,
since they both are right-angled triangles and have one common acute angle.


So, in the original right-angled triangle, the ratio of the longer leg to the shorter leg is 2, too.


ANSWER.  In the original triangle, the ratio of the longer leg to the shorter leg is 2.

Solved.



Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Draw right triangle ABC
I'll place point A as the 90 degree angle.
From point A, draw a perpendicular line to BC, which I'll label as "h".
Let D = base of the altitude on side BC

p,q = legs of the right triangle ABC
h = height or altitude perpendicular to BC
Hypotenuse BC is split into two pieces BD = x and DC = 4x
The ratio of those pieces is DC:BD = 4x:x = 4:1.

The goal is to find the ratio of p:q or q:p.
Your teacher wasn't clear about what the order should be.

Carefully note we have three right triangles:Furthermore, those right triangles are similar triangles.
You can use the angle angle (AA) similarity theorem to prove this claim.

The similar triangles then let us say the following:
BD/AD = AD/DC
x/h = h/(4x)
4x^2 = h^2
h^2 = 4x^2
We'll use this in a bit.


Now focus on right triangle DBA.
Use the Pythagorean theorem
leg^2 + leg^2 = hypotenuse^2
x^2+h^2 = p^2
x^2+4x^2 = p^2 ...... plug in h^2 = 4x^2
5x^2 = p^2
p^2 = 5x^2

Then focus on right triangle DAC.
Again you'll need to use the Pythagorean theorem
leg^2 + leg^2 = hypotenuse^2
(4x)^2+h^2 = q^2
16x^2+h^2 = q^2
16x^2+4x^2 = q^2 ...... plug in h^2 = 4x^2
q^2 = 20x^2

Now divide p^2 over q^2 to arrive at 1/4.
The x^2 terms conveniently cancel out.
The 5/20 reduces to 1/4.

We determined that (p^2)/(q^2) = (p/q)^2 = 1/4

Apply the square root to both sides to go from
(p/q)^2 = 1/4
to
p/q = 1/2
We'll only consider the positive result since side lengths cannot be negative, so I think it makes sense that the ratio shouldn't be negative either.

The ratio of p to q is 1:2
The ratio of q to p is 2:1
In short, one leg is twice as long as the other.

I used GeoGebra to confirm the answer is correct.

One example triangle you could form is to have the horizontal leg of 3 units and vertical leg of 6 units (feel free to pick other values; just make sure of course one leg is twice the other). Use GeoGebra or similar software tools to construct such a triangle. Then construct the altitude as shown above and note the ratio of the pieces of the hypotenuse. You should get 4:1 as that ratio.
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