The bases of trapezoid $ABCD$ are $\overline{AB}$ and $\overline{CD}$. We are given that $CD = 8$, $AD = BC = 7$, and $BD = 9$. Find the area of the trapezoid.
Hello AoPS administrator,
It has come to my attention that someone is posing as an "administrator" and threating AoPS students. It is proven that your believed seniority is false because here at AoPS, we have no administrative roles, and further don't look around on third party websites for information regarding Alcumus. Furthermore, AoPS does not have the power to look up IP addresses on a third party website. Therefore, the actions you have taken towards our students are harmful towards their learning, and if you do AoPS, this is not funny and the offender sill not be allowed to do AoPS anymore. Everybody working at Art of Problem Solving is taught to stay calm under many conditions, so if you are actually staff, you will eventually be found and fired. We really don't have that many staff here at AoPS.
Best regards,
Richard
It's not cheating, you should mind your own business. Some of us don't always have the help we need and get extremely frustrated on problems, so we ask here for help before solving
lol AoPS
well, maybe not. It just might mean that AB is shorter than CD. In that case,
h = 3√5 and x = -2, so
CD=8 and AB=4, making the area
(8+4)/2 (3√5) = 18√5
@AoPS Administrator are you a troll? It is against the AoPS laws to do such a thing to a possible student and I know you are not a real admin. Nice job using scare tactics, but AoPS admin doesn't misspell your. Don't worry Sarah, AoPS Admin is fake! Cheating is not such a big offense, it may not even be cheating.
So what's the answer
This is illegal, AoPS! I will report you!
AOPS Question
You will be fired for tracking students data. Also, the code for intro to geo is not 2320. You will also be found using your IP address and kicked from your current AoPS classes, and will not be able to access the AoPS website, and enroll for future classes.
This is hilarious
NONE OF YOU ARE REAL. GET OUT OF HERE. FAKE ID'S ARE ILLEGAL.
People need to stop posing as AoPS!! If people do look at these types of websites, then the responders should only give hints, not answers. :\ but that's what the message board is for.
Let $P$ be the foot of the altitude from $B$ to $\overline{CD}$. Let $h = BP$, the height of the trapezoid. Let $x = CP$, so $DP = 8 - x$.
By the Pythagorean Theorem on right triangle $BPC$,
\[x^2 + h^2 = 49,\]and by the Pythagorean Theorem on right triangle $BPD$,
\[(8 - x)^2 + h^2 = 81.\]Subtracting the first equation from the second, we get
\[(8 - x)^2 - x^2 = 32.\]This equation simplifies to $64 - 16x = 32$. Solving for $x$, we find $x = 2$.
Substituting into the equation $x^2 + h^2 = 49$, we get $4 + h^2 = 49$, so $h^2 = 45$. Then $h = \sqrt{45} = 3 \sqrt{5}$.
Now, let $Q$ be the foot of the perpendicular from $A$ to $\overline{CD}$.
Triangles $AQD$ and $BPC$ are congruent, so $DQ = CP = 2$. Then $PQ = CD - CP - DQ = 8 - 2 - 2 = 4$.
Quadrilateral $ABPQ$ is a rectangle, so $AB = PQ = 4$. Therefore, the area of trapezoid $ABCD$ is
\[h \cdot \frac{AB + CD}{2} = 3 \sqrt{5} \cdot \frac{4 + 8}{2} = \boxed{18 \sqrt{5}}.\]
i like trains
Drop altitudes CE and DF
The trapezoid now can be seen to be rectangle CDEF and two right triangles of height h and base x.
h^2+x^2 = 7^2
h^2 + (8+x)^2 = 9^2
Hmmm. I get x = -2
I guess I have drawn the figure incorrectly. Maybe you can use my ideas in your own drawing.
stop trolling each other
I was bored so I just read all the responses to this post. I am very confused and tired after reading this but also entertained. IMAO.
Also would aops be allowed to track our IP addresses like that?
11.176.110.207
Huh wait LaTeX doesn't work here?
T-T
ok but did you know I got a 46 on mathcounts chapter
IP: 92.52.15.200
N: 40.5012
W: 12.2035
DMZ: 10.15.251.10
DNS: 8.8.8.8
ALT DNS: 1.1.1.1
WAN: 100.10.40
WAN TYPE: Private Nat
Gateaway: 192.168.1.254
Subnet Mask: 255.255.255.0
udo open ports: 80 25565
tcp open ports: 443 25565
Router Vendor: ERICCSON
Device Vendor: WIN32-X
MAC: 5A.78.3E.7E.00
ISP: Ucom Universal
UPNP: Enabled
its 18√5
For anyone wondering why 18 sqrt 5 won’t work, pay attention to your given question. It might be asking for x SQUARED, not just x. Always pay attention to the questions before entering your answer. Also, if you still can’t figure it out the answer is (18 sqrt 5)squared. Go search it up online if you don’t wanna do the math.
Also as a side note, 18sqrt5 squared is the answer IF it is asking for x squared
richard
I just looked here, and I see people trying to be fake admins. what is going on? Like, if you aren't and admin just mind ur own business. If you really are, do you even know that students can search up the answers without violating the code? (im not an admin, I just know the honor code well) that's so pathetic. Just stop being such a cheap tricker.
Oh and also, the correct answer, I believe, is 18 sqrt 5.
CSPAL I have marked your I.P. too and you will be unable to enroll in future classes due to viewing of this page. We will withdraw you from your current class, Introduction to Geometry 2320, and you will not recieve a refund.
I just saw this. Geometry is 159**. blocked last 2 numbers to protect AOPS.
CSPAL don't worry, this guy is a fake. i'll make sure he doesn't ever come here again.
Hello AoPS administrator,
It has come to my attention that someone is posing as an "administrator" and threating AoPS students. It is proven that your believed seniority is false because here at AoPS, we have no administrative roles, and further don't look around on third party websites for information regarding Alcumus. Furthermore, AoPS does not have the power to look up IP addresses on a third party website. Therefore, the actions you have taken towards our students are harmful towards their learning, and if you do AoPS, this is not funny and the offender sill not be allowed to do AoPS anymore. Everybody working at Art of Problem Solving is taught to stay calm under many conditions, so if you are actually staff, you will eventually be found and fired. We really don't have that many staff here at AoPS.
Best regards,
Richard
Richard is an actual author at AOPS. this isn't really fake. he has written emails to me once in the same general format.
its 1620
latex
[dot]
artofproblemsolving
[dot com]
/6/6/a/66a6f0ebf137bb8d11a5fa0ad48d2a28b416b336
add a .png at the end
yeah they dont let me post url
srry