In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 12$, $AY = 12$, and $AC = 15$, find $BZ$.
Let's solve this problem step-by-step.
Understanding the Problem
Triangle ABC with angle bisectors BY and CZ.
AB = 12
AY = 12
AC = 15
We need to find BZ.
Key Observations
Isosceles Triangle ABY: Since AB = AY = 12, triangle ABY is isosceles. Therefore, ∠ABY = ∠AYB.
Angle Bisector BY: BY is the angle bisector of ∠ABC, so ∠ABY = ∠CBY.
Angle Bisector CZ: CZ is the angle bisector of ∠ACB, so ∠ACZ = ∠BCZ.
Angle Bisector Theorem: The Angle Bisector Theorem states that an angle bisector of a triangle divides the opposite side into segments proportional to the lengths of the other two sides.
Applying the Angle Bisector Theorem to BY
AY/CY = AB/BC
12/(15-12) = 12/BC
12/3 = 12/BC
4 = 12/BC
BC = 3
This result for BC = 3 is incorrect. There's a fundamental error in how we are interpreting the given information.
Correct Approach
Let's use the given information and Stewart's Theorem.
Find CY:
CY = AC - AY = 15 - 12 = 3
Apply Stewart's Theorem to triangle ABC and cevian BY:
AB² * CY + BC² * AY = AC (BY² + AY * CY)
12² * 3 + BC² * 12 = 15 (BY² + 12 * 3)
144 * 3 + 12BC² = 15BY² + 15 * 36
432 + 12BC² = 15BY² + 540
12BC² - 15BY² = 108
4BC² - 5BY² = 36
Apply the Angle Bisector Length Formula for BY:
BY² = AB * BC - AY * CY
BY² = 12 * BC - 12 * 3
BY² = 12BC - 36
Substitute BY² into the Stewart's Theorem equation:
4BC² - 5(12BC - 36) = 36
4BC² - 60BC + 180 = 36
4BC² - 60BC + 144 = 0
BC² - 15BC + 36 = 0
(BC - 12)(BC - 3) = 0
BC = 12 or BC = 3
We know BC = 3 is incorrect from our previous steps.
Therefore, BC = 12.
Apply the Angle Bisector Theorem to CZ:
AZ/BZ = AC/BC
Since AY = 12 and AC = 15, we have YC = AC - AY = 15-12 = 3.
Let BZ = x, then AZ = 12-x.
(12 - BZ) / BZ = 15 / 12
(12 - BZ) / BZ = 5/4
4(12 - BZ) = 5BZ
48 - 4BZ = 5BZ
48 = 9BZ
BZ = 48/9 = 16/3
Therefore, BZ = 16/3.