Answer:
The measure of DE is 12
Step-by-step explanation:
* Lets study the figure to solve the problem
- There are two intersected circles B and C
- BA is a radius of circle B and CE is a radius of circle C
- AD is a tangent to the two circles touch circle B in A and touch
circle C in E
- The line center BC intersects the tangent AD at D
- There are two triangles in the figure Δ BAD and Δ CED
* Now lets solve the problem
∵ AD is a tangent to circles B and C
∵ BA and CE are radii
∴ BA ⊥ AD at A
∴ CE ⊥ AD at E
- Two lines perpendicular to the same line, then the two lines are
parallel to each other
∴ BA // CE
- From the parallelism
∴ m∠ABD = m∠ECD ⇒ corresponding angles
∴ m∠BAD = m∠CED ⇒ corresponding angles
- In any two triangles if their angles are equal then the two triangles
are similar
- In the two triangles BAD and CED
∴ m∠ABD = m∠ECD ⇒ proved
∴ m∠BAD = m∠CED ⇒ proved
∵ ∠D is a common angle of the two triangle
∴ The two triangle are similar
- There are equal ratios between their sides
∴ BA/CE =AD/ED = BD/CD
∵ BD = 50 , AD = 40 , CD = 15
∴ 40/ED = 50/15 ⇒ using cross multiplication
∴ ED(50) = 15(40)
∴ 50 ED = 600 ⇒ divide both sides by 50
∴ ED = 12
