GMAT Club World Cup 2022 (DAY 5): In a quarter of a circle, a right an

Question

https://gmatclub.com/forum/download/file.php?id=67312 Right angle ABC is inscribed in a quarter of a circle, as shown above. What is the length of line segment DC ? A. 3 B. 4 C. 5 D. 7 E. 15 2022.png

Papai21 · Accepted Answer

Answer choice is C

ThatDudeKnows · Answer

Extend the figure to the right by drawing the other quarter of the circle and extending BC until it intersects with the circle at point E. AB is 24. 
Starting with triangle ABE, we have right triangle with a side that looks like it'll make a tidy 3-4-5. Let's see if we can backdoor our way to the solution.
AB is 24 and is the shortest side of triangle ABE. If that is the "3" in a 3-4-5, the other leg will be 32 and the hypotenuse will be 40. The hypotenuse is 2r, so r=20. We are given that BC is 7, so that would make CE 25 and CO is 15. That makes DC = 5.
ABE and COE are similar and ABE is a 3-4-5, so COE should be, too. It is, so everything checks out.

Answer choice C.

paragamrutkar · Answer

Screenshot 2022-07-15 at 10.19.39 PM.png Steps - 1. Find AC using the Pythagorean theorem. AC = 25 2. riangle ACE is an isosceles triangle. Therefore, CE = AC = 25 . 3. BE = 25 + 7 = 32 6. From Right angled riangle ABE , AE = 40. Hence, Radius of Circle is 20. 7. From Right angled riangle AOC , CO = 15. Hence, DC = 20 - 15 = 5 . Hence, the answer is C.

gmatophobia · Answer

Few inferences that we can make -

1) As  \angle ABC is 90 , if we extend BC to form BX, AX will be the diameter of the circle as shown in figure attached.
2) Also, the triangles formed  	riangle XOC  is similar to  	riangle XBA 
3) CA = CX

CA = 25 (we can use triplet 7 - 24 - 25)

Therefore CX = 25
BX = 7+25 = 32

AX^2  =  AB^2  +  BC^2

AX^2  =  24^2  +  32^2

AX^2  = 576 + 1024 = 1600

AX = 40

AO = OX = 20

Using similarity of triangle

\frac{OX}{XB}  =  \frac{CO}{BA}

CO =  \frac{OX * BA }{XB}

CO =  \frac{20 * 24}{32}

CO = 15

DC = 20 - 15 = 05

IMO C

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GMAT Club World Cup 2022 (DAY 5): In a quarter of a circle, a right an

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