Express x in terms of the other variables in the picture.?

Question

Im having trouble solving this problem... I've been trying to solve it for the past two days but i simple cant. Please help me.

Express x in terms of the other variables in the picture.

to see the picture go to

http://www.math.unb.ca/ready/paper/img118.gif



Please provide an explanation on how to solve the problem. Thank You.

Answer 1

a.
r/h = (x+t)/t
Add together the variables in the bottom to create the length for the larger triangle.
They are similar triangles.
So their sides are similar.
Ratios [r : h] and [(x+t) : t]


Cross multiply, distribute h and then solve for x.
rt = h(x+t)
rt = hx + ht
rt - ht = hx
(rt - ht) / h = x <-- You can factor if you prefer to:
t(r-h) / h = x

b. Without trying to add variables here...
large triangle : small triangle
[?² + t² = x²] : [h² + ?² = r²]

So, solve for the large triangle side that is unknown.
?² + t² = x²
?² = x² - t²
? = √(x² - t²)
This is your new side for the larger triangle that is proportional to h.

Now, do ratios.
[(√(x² - t²)) : h] and [x : r]

Cross multiply and solve for x.
√(x² - t²) / h = x / r
r(√(x² - t²)) = hx
[r(√(x² - t²))] / h = x

PS This can be done multiples of ways. Just do it how it makes sense in your head.

Answer 2

Just set up proportions from the similar triangles and solve for x. In both cases, there is a smaller triangle inside a larger triangle where you can set up relations. I'm sure you'll understand what I mean once you have my work and the pictures in front of you.

1. In this one, the legs of the smaller triangle are proportional to each other in the same way that the legs are in the larger one. Mathematically, we have:

h / t = r / (t + x)
==> cross multiply
h∙(t+x) = t∙r
==> distribute on left side
h∙t + h∙x = t∙r
==> subtract h∙t on both sides
h∙x = t∙r - h∙t
==> factor t on right side
h∙x = t∙(r - h)
==> divide by h
x = t∙(r - h) / h ... ANSWER

2. In this one, the legs are proportional to the hypotenuses in the same way for the small and larger triangles, so we can set up this proportion and solve for x, although we have a side missing. Therefore, we will need to use the Pythagorean Theorem to find the third side of the small triangle, which we'll call a:

h² + a² = r²
==> subtract h² on both sides
a² = r² - h²
==> square root both sides
a = √(r² - h²)

So, we can plug this in and set up the following proportion and solve for x:

x / t = r / √(r² - h²)
==> multiply by t on both sides
x = (r∙t) / √(r² - h²) ... ANSWER

Answer 3

I'll do the first one...

Do you see two triangles in the diagram (a)? One small one and the larger one?

Since they share all angles, they are same triangles in different scale. Therefore, you can say:

r/(x+t) = h/t

You cross multiply and get:
rt=h(x+t)
Expand and get:
rt=hx+ht
Rearrange and get:
hx=rt-ht
divide both sides by h and you get:
x=(rt-ht)/h
factor and you get:
x=(t(r-h))/h

There you have it....

Hint for the (b)
Notice they didn't give you the same sides as variables, but you have two similar triangles. You can easily arrive at the missing variable by using a Pythagorean theorem.