Question

Pentagon RSTUV is circumscribed about a circle. What is the value of x if RS = 10, ST = 13, TU = 11, UV = 12, and VR = 12?

Answer 1

Answer:

Step-by-step explanation:

The value of x is 7 ⇒ 1st answer

Step-by-step explanation:

* Lets revise a fact in the circle

- The two tangents drawn from a point out side the circle are equal

∵ RSTUV is circumscribed about a circle

∴ Each side of the pentagon is a tangent to the circle

- Look to the attached figure to know how we will solve the problem

- Each tangent divided into two parts

# RS = x + y

∵ RS = 8

∴ x + y = 8 ⇒ (1)

# RV = x + n

∵ RV = 12

∴ x + n = 12 ⇒ (2)

- Subtract (2) from (1)

∴ y - n = -4 ⇒ (3)

# ST = y + z

∵ ST = 12

∴ y + z = 12 ⇒ (4)

# TU = z + m

∵ TU = 15

∴ z + m = 15 ⇒ (5)

- Subtract (5) from (4)

∴ y - m = -3 ⇒ (6)

# UV = m + n

∵ UV = 9

∴ m + n = 9 ⇒ (7)

- Add (6) and (7)

∴ y + n = 6 ⇒ (8)

- Lets solve equation (3) and equation (8) to find y

∵ y - n = -4 ⇒ (3)

∵ y + n = 6 ⇒ (8)

- Add (3) and (8)

∴ 2y = 2 ⇒ divide two sises by 2

∴ y = 1

- Lets substitute the value of y in equation (1)

∵ x + y = 8 ⇒ (1)

∵ y = 1

∴ x + 1 = 8 ⇒ subtract (1) from both sides

∴ x = 7

* The value of x is 7

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Answer 2

Answer:

x=6

y=7

z=4

a=8

b=4

Step-by-step explanation:

Let A,B,C,D,E are points on the circle or where RS,ST,TU,UV,VR touches the circle respectively.

RS=RAS=RA+AS=b+x=10

ST=SBT=SB+BT=x+y=13

TU=TCU=TC+CU=y+z=11

UV=UDV=UD +DV=z+a=12

VR=VER= VE +ER=a+b=12

∵

[AS=SB=x

BT=TC=y

CU=UD=z

DV=VE=a

ER=RA=b]

adding downwards

2(x+y+z+a+b)=10+13+11+12+12=58

x+y+z+a+b=58/2=29

(x+y)+(z+a)+b=29

13+12+b=29

b=29-25=4

b=4

4+x=10

x=6

6+y=13

y=13-6=7

7+z=11

z=11-7=4

4+a=12

a=12-4=8