The perimeter of a right triangle is 60 cm and the altitude perpendicular to the hypotenuse is 12 cm. What are the lengths of the three sides?

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- Aug 6th 2009, 07:44 PMConfused25length of three sides
The perimeter of a right triangle is 60 cm and the altitude perpendicular to the hypotenuse is 12 cm. What are the lengths of the three sides?

- Aug 6th 2009, 09:13 PMANDS!
I'm assuming altitude means height. If that is so, what do we know:

We know the perimiter of the triangle is 60cm, which means we know that , where B is the base, A is the altitude, and H is the hypotenuse of our triangle.

We also know that the Altitude (A) is 12CM, so:

Now we also know that by the Pytha Theorem yes?

Using these two formulas, we can do some substitutions and solve for these worrisome sides!

Can you take it from here? - Aug 6th 2009, 10:06 PMWilmer
- Aug 6th 2009, 10:14 PMANDS!
Well good show. Even easier. We've two right triangles formed by splitting an angle of 90 degrees into two. From there its simply a matter of using properties of a 45-45-90 triangle to figure out the sides.

My bad. - Aug 6th 2009, 10:30 PMWilmer
- Aug 6th 2009, 10:43 PMWilmer
OK; works like this (not as easy as it looks!):

Let a = one leg, b = other leg, c = hypotenuse

Since perimeter = 60, then c = 60 - a - b; so:

a^2 + b^2 = (60 - a - b)^2 ; simplify to get:

b = (60a - 1800) / (a - 60) [1]

Using areas: ab = 12(60 - a - b) ; simplify to get:

b = (720 - 12a) / (a + 12) [2]

From [1] and [2]:

(60a - 1800) / (a - 60) = (720 - 12a) / (a + 12) ; simplify to get:

a^2 - 35a + 300 = 0

(a - 20)(a - 15) = 0

a = 20 or 15 : that's the lengths of the 2 legs.

So hypotenuse = 60 - 15 - 20 = 25.

So you end up with a 15-20-25 right triangle. - Aug 7th 2009, 07:19 AMSoroban
Hello, Confused25!

Quote:

The perimeter of a right triangle is 60 cm and the altitude to the hypotenuse is 12 cm.

What are the lengths of the three sides?

Code:`C`

*

*| *

b * | * a

* |12 *

* | *

* | *

A *-----+-----------------* B

: - - - - - c - - - - - :

Given: right triangle

We have: .

From [1], we have: .

. .

Substitute into [1]: . .[4]

Substitute into [2]: . .[5]

Substitute [4] into [5]: .

Quadratic Formula: .

Hence: .

Substitute into [4]: .

- Aug 7th 2009, 07:34 AMmalaygoel
- Aug 7th 2009, 09:39 AMWilmer
I think Mr Soroban's having a bad day (Crying)

Right triangle 3-4-5 has height to hypotenuse = 1.2 is something

we memorize in grade 9 or so.

Anyhow, answer is a = 15, b = 20, c = 25.

As a general case, given p = perimeter and h = height to hypotenuse:

b,a = [-v +,- SQRT(v^2 - 4uw)] / (2u), where:

u = 2(p + h)

v = -p(p + 2h)

w = p^2 h

OK, Mr.Confused; now try this one:

perimeter = 1400, height to hypotenuse = 168 ; calculate a and b