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.

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1. why is this in LaTex lol

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2. 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.

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3. 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

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4. Steve, you solved the system of equations wrong. It is in fact 8-x, not 8+x.

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5. Stop cheating!

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6. lol AoPS

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7. AOPS Question

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8. 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

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9. We have marked youe I.P., and you will be unable to enroll for future AoPS courses.

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10. @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.

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11. 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.

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12. This is illegal, AoPS! I will report you!

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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

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14. 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.

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15. stop trolling each other

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16. NONE OF YOU ARE REAL. GET OUT OF HERE. FAKE ID'S ARE ILLEGAL.

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18. This is hilarious

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19. 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.

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20. 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
$h \cdot \frac{AB + CD}{2} = 3 \sqrt{5} \cdot \frac{4 + 8}{2} = \boxed{18 \sqrt{5}}.$

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21. Huh wait LaTeX doesn't work here?

T-T

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22. hi the answer is 35 sqrt 2

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23. 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?

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26. i like trains

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27. latex
[dot]
artofproblemsolving
[dot com]

add a .png at the end

yeah they dont let me post url
srry

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28. ok but did you know I got a 46 on mathcounts chapter

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29. its 18√5

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