A circle is circumscribed by a pentagon. The sides are all tangent to the circle. (It's a bit hard to explain without a picture, so I'll do my best.)
The pentagon is QSUWY. However, there are points in between all five sides. In SQ lies point R, SU point T, UW point V, WY point X, and QY with point Z.
It is given that QZ=9, YX=7, XW=12, UW=15, and SU=16. Find the perimeter of the pentagon. In all, I know the perimeter so far is 59, and the problem is I'm not sure how to solve for SQ and ZY. Help please? It's much appreciated.
Even though you did not state it, I will assume that the points R,T,V... are the points of contact of the sides with the circle.
One of the basic properties of tangents and circles is that
If tangents are drawn to a circle from an external point P, and those tangents touch at A and B, then PA = PB
You have that situation 5 times in your figure.
so QZ = QR, YZ= YX, etc.
work your way all around, you will be able to find each segment of the pentagon.
I got 88, but check my arithmetic, I am terrible in arithmetic.
To solve for SQ and ZY, we can use the concept of tangents to a circle and the properties of a pentagon. Let's start by analyzing the given information and establishing some relationships between the sides and diagonals of the pentagon.
1. Tangent segments from an external point to a circle are equal in length. Since ZQ and ZY are tangent segments to the circle, we can conclude that ZQ = ZY.
2. In a regular pentagon, all sides and diagonals are equal in length. However, the given pentagon QSUWY is not necessarily a regular pentagon. Therefore, we need to find another approach to solve for the missing side lengths.
To find the missing side lengths, we can use the fact that the sum of the lengths of opposite sides of a tangential quadrilateral (such as QSUWY) are equal. Let's consider the diagonals SQ and ZY as the opposite sides.
1. Opposite sides of the quadrilateral QSUWY: SQ and ZY
The given lengths: QZ = 9 and YX = 7
Using the property mentioned above, we can set up the following equation:
SQ + ZY = QZ + YX
SQ + ZY = 9 + 7
SQ + ZY = 16
2. Now, we can calculate the perimeter of the pentagon QSUWY using the given lengths and the derived length from the equation above.
Perimeter = SQ + SU + UW + WY + YX
= SQ + 16 + 15 + 12 + 7
Since SQ + ZY = 16, we can substitute SQ with (16 - ZY) in the above equation to express the perimeter in terms of ZY.
Perimeter = (16 - ZY) + 16 + 15 + 12 + 7
= 56 + (16 - ZY)
Given that the perimeter is already determined to be 59, we can now solve for ZY.
56 + (16 - ZY) = 59
56 + 16 - ZY = 59
72 - ZY = 59
-ZY = 59 - 72
-ZY = -13
ZY = 13
Now that we know ZY = 13, we can substitute this value into the perimeter equation to find the missing side length SQ.
Perimeter = (16 - ZY) + 16 + 15 + 12 + 7
= (16 - 13) + 16 + 15 + 12 + 7
= 29 + 16 + 15 + 12 + 7
= 79
Therefore, the perimeter of the pentagon QSUWY is 79.