If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Challenge problems: circumscribing shapes

Solve two challenging problems that apply properties of tangents to find the perimeter of a circumscribing shape.

## Problem 1

All sides of $\mathrm{△}ABC$ are tangent to circle $P$.
What is the perimeter of triangle $ABC$?
units

## Problem 2

All sides of quadrilateral $ABCD$ are tangent to circle $P$.
What is the perimeter of quadrilateral $ABCD$?
units

## Want to join the conversation?

• I am confused. How did you solve when you still haven't found x?
• In both cases, the x cancels itself out. Take question 2: the perimeter is a whole bunch of numbers which I am too lazy to type out, and then there is an unknown length which I will call x. To find the other unknown length, you can just take 12-x. The perimeter will be all those numbers + x + 12 - x. The solution to x+12-x is 12, as you can rearrange it as x-x+12.
• what are the answer to number 1 and/or 2. Im very confused on this subject matter.
• Hi! I think I might be able to help :)

So basically for #1, the main theorem you want to apply to the problem is the theorem that "2 tangent lines drawn to a circle from the same point are congruent." Let's call the point between AC "x", the point between BC "y" and the point between AB "z".
Because of the theorem, you can say that YB is congruent to ZB (YB = ZB = 16). You have one piece of the puzzle right there!

Now you can move on to the other pairs of tangent lines. Based on the same theorem we can say that XC is congruent to YC, and that XA is congruent to ZA. Let's set XC equal to "x" (XC = YC = "x") Because AC is equal to 14, we can call XA "14-x." (XA = ZA = "14-x")

Now, let's add it all up! 2x + 2(14-x) + 2(16) = 28 + 32 = 60.

#2 is extremely similar. If you want more help, don't hesitate to ask me or anyone else on here! :)
• i did not understand the way to find perimiter of quadrilaterals and triangles
• The perimeter of a shape is just the sum of the lengths of all the sides. So for a triangle you add up all 3 sides and for a quadrilateral you add up all 4 sides.
• still don't get it
• Challenge problems: circumscribing shapes Problem 1
What logic tells us that we can use the same variable x on line AB versus line AC? They visually look to be very different lengths.
• we do not have to consider AB=AC=x, just considering each segment as an individual pair of segments, for example, DC= x+y and BC=y+z and AB=m+z (by the way z=9.6 and m=3.7)...AD=m+x..........
• makes no since