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Point I is the incenter of triangle ABC in the figure. The figure shows AB = 10, AC = 8, ∠A

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Point I is the incenter of triangle ABC in the figure. The figure shows AB = 10, AC = 8, ∠A

by Max@Math Revolution » Mon Jan 27, 2020 6:21 pm
[GMAT math practice question]

Point I is the incenter of triangle ABC in the figure. The figure shows AB = 10, AC = 8, ∠ABI = 22° and ∠ACI = 30°. Line DE is parallel to line BC. What is the length of the perimeter of triangle ADE?
1.27PS.png
A. 14
B. 16
C. 18
D. 20
E. 22
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Re: Point I is the incenter of triangle ABC in the figure. The figure shows AB = 10, AC = 8, ∠A

by Max@Math Revolution » Tue Jan 28, 2020 11:51 pm
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Since I is the incenter of the triangle ABC, IB and IC bisect ∠ABC and ∠ACB, respectively. We know ∠IBC = 22° and ∠ICB = 30°. Since ∠DIB and ∠CBI are alternate interior angles, we have ∠BID = 22° and triangle DBI is an isosceles. Since ∠EIC and ∠BCI are alternate interior angles, we have ∠EIC = 30° and triangle CEI is an isosceles. Thus we have the perimeter of triangle ADE = AD + DE + EA = AD + DI + IE + EA = AD + DB + CE + EA = AB + AC = 10 + 8 = 18.

Therefore, C is the answer.
Answer: C
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