Math Help - [SOLVED] SAT geometry

In the figure below, arc SBT is one quarter of a circle with center R and radius 6. If the length plus the width of rectangle ABCR is 8, then the perimeter of the darkened region is

a)

b)

c)

d)

e)

Can you please show the steps? Thanks in advance

hey i forgot a major info. The graph stated that SR is 6, sorry i didn't include it on the graph!!

Originally Posted by fabxx

In the figure below, arc SBT is one quarter of a circle with center R and radius 6. If the length plus the width of rectangle ABCR is 8, then the perimeter of the darkened region is

a)

b)

c)

d)

e)

Can you please show the steps? Thanks in advance

i will start you off.

Let be , that is, the width of the rectangle.
Let be , that is, the length of the rectangle.
Let be the perimeter of the darkened region.

Note that the diagonal

we are given ............(1)

by Pythagoras' theorem, .............(2)

you have two simultaneous equations with two unknowns, you can solve for and . the rest should be straight forward



now,





, and you know .

so you can find all the components and hence the perimeter

Hey i forgot to include a major info in the graph. The radius is 6 is meant that SR is 6. I already corrected it in my previous thread. (:

Originally Posted by fabxx

Hey i forgot to include a major info in the graph. The radius is 6 is meant that SR is 6. I already corrected it in my previous thread. (:

as you can see, we already figured that out. you told us that R was the center and the circle had a radius of 6 (RT = 6 also ). nothing changes in what i did

Fabxx,

This is SATs. Pythagoras is too long for SAT. You are supposed to get this answer in not more than 30 seconds, without a calculator. The worst thing a candidate make in a SAT exam is take it with a Texas Instruments. Really, this is not a joke.

To solve this question, follow me carefully:
It is given that AR + RC =8 ...(1)
SR=RB=CA=6 ...(2)

Perimeter = SA + arc ST + TC + CA (From above, we already know CA=6)

SA = 6-AR
TC= 6 - RC

Add the two equations above:
SA + TC = 12 - (AR+RC) = 12 - 8 =4

Therefore,

Perimeter = arc ST + 4 + 6 = + 4 +6 = 10+3


As you can see, you save time by not solving simultaneous equations and not evaluating pythagoras expression.

Hope that helps.

Originally Posted by shailen.sobhee

Fabxx,


Perimeter = arc ST + 4 + 6 = + 4 +6 = 10+3

how did you get that arc ST is ?

Thanks in advance (:

Originally Posted by fabxx

how did you get that arc ST is ?

Thanks in advance (:

gives the circumference. you have 1/4 of it (it's a quarter circle)

Thanks to shailen.sobhee and Jhevon!

Originally Posted by shailen.sobhee

Fabxx,

SR=RB=CA=6 ...(2)

How did you get that CA=6? CA isn't the radius, and from the graph, you can easily see that CA≠SR

Thanks in advance

Originally Posted by fabxx

How did you get that CA=6? CA isn't the radius, and from the graph, you can easily see that CA≠SR

Thanks in advance

you are correct. but RB is the radius, and CA = RB since they are both diagonals of the rectangle (sobhee mentioned this)