So this question is for my Trig homework and I'm extremely confused on it. Some help please?

Question

The hypotenuse of a right triangle is 20cm. Express the area of the triangle as a function of the length x of one of the legs.

Answer 1

First use the Pythagorean Theorem.

The hypotenuse is 20. One of the legs has length x. Call the length of the other leg y.

x^2 + y^2 = 20^2
x^2 + y^2 = 400

Now solve for y in terms of x.

y^2 = 400 - x^2
y = sqrt(400 - x^2)

Now, since the triangle is right, the two legs are the altitude (or height) and the base.

A = (1/2)*b*h
A = (1/2)*x*y
A = (1/2)*x*sqrt(400-x^2)

There's the area in terms of x!

Answer 2

I'm a little rusty but I do know that the area of a right triangle 1/2 the base times the height ( A=1/2 a*b). However, since the legs re not known you must figure those out. I suppose that's where the function of x comes in. There are 3 different types of right triangles which would determine figuring out the bases as the formulas are different. In a general right triangle the sides are as follows: c=the square root of "a" squared + "b" squared. That would mean that 20= the square root of "a" squared + "b" squared. Hmmm, you would have 2 unknowns and thus have x and y in trying to solve for functions. I'm afraid that's all I know for certain. You can always look up more information at the site listed below for a crash course.

Answer 3

The area of a right tringle is half the product of the legs.
One leg has length x. The other leg has length y.

x^2 + y^2 = 20^2 = 400
y^2 = 400 - x^2
y = sqrt(400-x^2)

So the area is (1/2)xsqrt(400-x^2)

Answer 4

let
z=20(hypotenuse)
x= base
y=height

by the formula of area of a triangle:

A= 1/2(bh) area is half of the product of the base and the height
or
A=0.5bh

now the hypotenuse is given to be 20.

Using the formula x^2 + y^2 =z^2 (Pythagorean)

x^2 + y^2 =20^2 (substitute value for z)

x^2 = y^2-400

y^2 = 400-x^2 ( solving for y)

Using the formula for area;

A= 0.5bh

let

b= base =x

h=height=y

A=x[( 400-x^2)]^1/2

( simply put its ' x ' multiply to the square root of 400-x ^2)


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its somewhere here, the area is a function of one of its legs x.
something like this, correct me if i'm wrong.

Answer 5

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Answer 6

Well, the area of the triangle is 0.5xy, where x and y are the two legs. And you know that x^2 + y^2 = 20^2, so y = sqrt(400-x^2), and the area is 0.5x*sqrt(400-x^2).