∠ In a Δ XYZ, XT is the bisector of X meeting side YZ at T. If XY = 5.6 cm, YZ = 6 cm, and YT = 3.2 cm, find XZ.
4.9 cm
This problem is a direct application of the Angle Bisector Theorem. The theorem states that when a bisector is drawn from a vertex of a triangle to the opposite side, it divides that opposite side into two segments that are proportional to the other two sides of the triangle. In simple terms, the bisector splits the base in the same ratio as the two arms enclosing the bisected angle.
Here XT bisects ∠X and meets YZ at T, so the theorem gives:
YT / TZ = XY / XZ
We are told XY = 5.6 cm, YZ = 6 cm and YT = 3.2 cm. Since T lies on YZ, the second segment is:
- TZ = YZ − YT = 6 − 3.2 = 2.8 cm
Substituting into the proportion and letting XZ = x:
- 3.2 / 2.8 = 5.6 / x
- Cross-multiplying: 3.2 × x = 5.6 × 2.8
- 3.2x = 15.68
- x = 15.68 ÷ 3.2 = 4.9 cm
So XZ = 4.9 cm. A quick sanity check confirms the logic: since YT (3.2) is a little larger than TZ (2.8), the side XY (5.6) opposite the shorter conception must be larger than XZ — and indeed 5.6 > 4.9, matching the proportional split.
The value 6 cm is merely the full length of YZ and cannot itself be XZ. Rounded guesses such as 5.4 cm or 4.2 cm do not satisfy the exact ratio 3.2 : 2.8; only 4.9 cm makes the proportion balance.
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