NeoPets Riddles (Lenny Conundrums) and Answers Here

Let us consider triangle ABC with sidelengths BC = a, AC = b, AB = c
and center of the inscribed circle (incenter) I. Drop perpendicular
altitudes from I to Ia, Ib and Ic on the sides of ABC. The lengths of
these altitudes are equal to the radius r of the incircle. This radius
we will have to calculate in order to find the area of the inscribed
circle (= incircle).

C
/ \
/ \
/ \
Ib Ia
/ \
/ I \
/ \
A-------Ic------B


First, we know that area(ABC) = area(AIB) + area(BIC) + area(CIA).

We can calculate area(AIB) = 0.5 * AB * IIc = 0.5cr. And in the same
way we find area(BIC) = 0.5ar and area(CIA) = 0.5br.

The conclusion is that area(ABC) = 0.5(a+b+c)r = sr, where s is the
semiperimeter of ABC.

Then, from Heron's formula, we also know that:

area(ABC) = sqrt(s(s-a)(s-b)(s-c))

Combining the two formulas for the area of ABC we can derive:

sr = sqrt(s(s-a)(s-b)(s-c))
r = sqrt(s(s-a)(s-b)(s-c)) / s
r = sqrt((s-a)(s-b)(s-c) / s )

And now we can finish off the area of the incircle:

area(incircle) = pi * r^2
= pi * sqrt((s-a)(s-b)(s-c) / s )^2

pi(s-a)(s-b)(s-c)
= -----------------
s

I most deffinately understood nothing in that post.

If my math isn't too rusty, I found the radius to be ~76, and we can form a right triangle towards that corner using the 76 as two side lengths, and then determine the third side length, the hypotenuse to be ~107. And since I used a right triangle, that means that the angle of the circle inscribe is obviously 90 degrees. I just cant remember how to determine area of a circular base, anyone?

I submitted my answer a few minutes ago, hope I can get it right:P

Is anyone going to share tonight?? I just simply cannot work it out. I know we have to use last week's answer to help work it out. I am not good with geometry or trig. please I could use some help.

I will be glad to help when it is another subject, something i am better at than geometry. I have been out of school now for almost 30 years. Thanks so much for helping, I will be glad to help another week when I can and post the answer then to help out someone else.

Thank you so much!! ~T.

i'm currently working on it and will soon be posting my answer...
don't get to get on as early as i use to due to a change in work schedule

This is what I have so far if it helps anyone:

s = (a+b+c)/2
= (330+240+270)/2
= 420

K= sqrt s(s-a)(s-b)(s-c)
= sqrt 420(90*180*150)
= 31946.83

r=K/s
=76.06388
Area of circle using Heron's formula = 18181.81 (from last week)
Confirmation
Area of triangle = r*s
= 76.06388 * 420
= 31496.83

To be continued....

any1 know the answer to the lenny? i'm so bad at trig... really
help!!!

owlette you have transposed your numbers on the confirmation part of your equation.

the number should also be:

76.06388 * 420 = 31946.83


"you have the 4 and the 9 transposed in the confirmation area of your work."

Have you any luck with the rest of the answer?? Just wondering. Thanks for your hard work and for posting your work so far. I just didn't want people getting confused with the transposed numbers.

~Tami

my solution is a little long to explain....
but i got the total area of the two triangles above is 7448.6722
the area of the blue sector is:
99.592/360 * pi * r^2
0.27664 * 3.1415 * 5785.1236
5027.7267

then subtract to get the red shaded area
7448.6722 - 5027.7267 = 2420.9455

Round up to nearest 25

2425

now i'm going to bed, good night all, it's about 4:30 am here

I think this thread is only for the Lenny Conundrum, so its not very likely that someone would submit an answer for the mystery pictures. You too, aznneopetsfan.

I think this is what most people do and also kind of the point of this topic. However, it isnt right to keep asking for answers and 'bludging', or flooding the topic with unnecessary posts.

I'm stuck on this week's Lenny Conundrum so I'll just go with Roly_Poly_Sandwiches' answer. The diagram looks promising and I kind of understand most of the formulae, so thanks for the answer =).

sure..... you would think


except that i've seen more mystery pic answers in this thread than in any thread supposedly dedicated to the mystery pic :p

From what i remembered,
our teacher taught us that in such situation,
AM=BM(points shown in the diagram).
So i was just wondering how you got AM=92.4652, BM=104.8
I'm not saying that I'm 100% right,
but can you please check your workings and see if there's any mistakes.
thank you very much:)

the board will be messed up if someone posts the answer to MC as many people are attempting to figure out LC at the moment? so i reckon it's better to stick with LC for now.

you're right...from a geometry site:
"Two tangent segments to a circle from an external point are congruent."




same site... http://agutie.homestead.com/files/heron/


states as fact that:



which is very helpful

.

im new here and this is the first LC i tried with this site and i will try to help if i can.............unless its really complicated for me like this one is

R = 76.06388
Per mollymil's find (good find!), the two segments that are tangent are 90.
Therefore, the area of the kite is 6845.7 (R*90).

There are several ways to find the angle of the sector of the circle.
It is 99.594.
That makes the area of the sector 5028.5 (circle area * 99.594/360).

Therefore, the area in question is 1817.2.
Rounded up to the nearest 25 it is 1825.

ummm... an answer anyone, please? I'm kinda getting confused