Question

1. c is the incenter of δxyz. find the lengths indicated.

zl =
cz =
ly =
cn =
cy =
cm =
khoa pham © 2025

Explanation:

Step1: Recall Incenter Properties

The incenter \( C \) of a triangle is equidistant from all sides, so \( CN = CL = CM \). Also, tangents from a point to a circle (incircle) are equal, so \( ZL = ZN \), \( XL = XM \), \( YM = YN \)? Wait, no, for \( \triangle XYZ \), the incenter \( C \) has perpendiculars to sides: \( CN \perp XZ \), \( CL \perp YZ \), \( CM \perp XY \), so \( CN = CL = CM \). Also, the segments from vertices to points of tangency: \( ZN = ZL \), \( XN = XM \), \( YM = YL \)? Wait, let's check the given lengths. \( XZ = ZN + NX = 38 \), \( NX = 18 \), so \( ZN = 38 - 18 = 20 \)? Wait, no, the diagram: \( ZX \) is 38, \( NX \) is 18, so \( ZN = ZX - NX = 38 - 18 = 20 \)? Wait, but maybe \( ZL = ZN \) (tangents from \( Z \) to incircle), so \( ZL = ZN \). Wait, \( XM = XN = 18 \) (tangents from \( X \) to incircle). \( XY = 57 \), so \( YM = XY - XM = 57 - 18 = 39 \), so \( YL = YM = 39 \) (tangents from \( Y \) to incircle). Then \( ZL \): \( ZY \) is? Wait, maybe I misread. Wait, the incenter's perpendiculars: \( CN \) is perpendicular to \( XZ \), length \( CN \): since \( CN \) is a perpendicular from incenter to \( XZ \), and maybe \( CN \) is equal to \( CL \) and \( CM \). Wait, the problem has \( CN \) written as \( 18^2 \)? No, maybe that's a typo. Wait, let's re-express:

1. \( ZL \): Tangents from \( Z \) to incircle: \( ZL = ZN \). \( ZN = ZX - XN \). \( ZX = 38 \), \( XN = 18 \), so \( ZN = 38 - 18 = 20 \)? Wait, no, maybe \( XN = 18 \), so \( ZN = ZX - XN = 38 - 18 = 20 \), so \( ZL = ZN = 20 \)? Wait, but the other side: \( XY = 57 \), \( XM = XN = 18 \), so \( YM = 57 - 18 = 39 \), so \( YL = YM = 39 \) (tangents from \( Y \) to incircle). Then \( ZL \): if \( ZL = ZN \), and \( ZN = 38 - 18 = 20 \), so \( ZL = 20 \). Wait, but maybe the given \( CN \) is equal to \( CM \) and \( CL \). Let's assume \( CN = 18 \) (maybe the \( 18^2 \) is a mistake). So \( CN = CL = CM = 18 \).

2. \( CZ \): To find \( CZ \), use Pythagoras in \( \triangle CZN \): \( CZ = \sqrt{ZN^2 + CN^2} \). \( ZN = 20 \), \( CN = 18 \), so \( CZ = \sqrt{20^2 + 18^2} = \sqrt{400 + 324} = \sqrt{724} \)? No, that can't be. Wait, maybe \( ZN = 19 \)? Wait, the diagram has \( ZN \) labeled 19? Wait, the user's diagram: \( Z \) to \( N \) is 19? Wait, the original problem: "Z" to "N" is 19? Wait, the user's image: "Z" to "N" is 19, "N" to "X" is 18, "X" to "M" is 18, "X" to "Y" is 57. So:

- Tangents from \( X \): \( XN = XM = 18 \) (since \( CN \perp XZ \), \( CM \perp XY \), so \( XN = XM \) as tangents from \( X \) to incircle).

- Tangents from \( Z \): \( ZN = ZL \). \( ZN = 19 \)? Wait, \( ZX = ZN + NX = 19 + 18 = 37 \)? But the diagram says \( ZX \) is 38. Maybe a typo. Wait, let's take the given numbers:

\( ZX = 38 \), \( NX = 18 \), so \( ZN = 38 - 18 = 20 \).

\( XY = 57 \), \( XM = 18 \), so \( YM = 57 - 18 = 39 \).

Thus:

- \( ZL = ZN = 20 \) (tangents from \( Z \) to incircle: \( ZL \) and \( ZN \) are both tangents, so equal).

- \( LY = YM = 39 \) (tangents from \( Y \) to incircle: \( LY \) and \( YM \) are equal).

- \( CN = CL = CM \): since \( CN \) is perpendicular to \( XZ \), \( CL \) to \( YZ \), \( CM \) to \( XY \), so they are equal (inradius). Let's assume \( CN = 18 \) (from the diagram's \( NX = 18 \), maybe \( CN \) is 18? Wait, the handwritten \( CN: 18^2 \) is probably a mistake, maybe \( CN = 18 \).

- \( CZ \): In \( \triangle CZN \), \( ZN = 20 \), \( CN = 18 \), so \( CZ = \sqrt{ZN^2 + CN^2} = \sqrt{20^2 + 18^2} = \sqrt{400 + 324} = \sqrt{724} \approx 26.9 \)? No, that doesn't make sense. Wait, maybe \( ZN = 19 \) (as per diagram's label "19" near \( ZN \)). So \( ZN = 19 \), \( NX = 18 \), so \( ZX = 19 + 18 = 37 \), but diagram says 38. Maybe the diagram has \( ZX = 38 \), \( ZN = 20 \), \( NX = 18 \) (20 + 18 = 38). Then \( ZL = ZN = 20 \).

- \( CN \): Since \( CN \) is perpendicular to \( XZ \), and if \( CN \) is equal to \( CM \) and \( CL \), and \( XM = 18 \), \( XY = 57 \), so \( YM = 57 - 18 = 39 \). Then \( CY \): in \( \triangle CYM \), \( YM = 39 \), \( CM = CN = 18 \), so \( CY = \sqrt{39^2 + 18^2} = \sqrt{1521 + 324} = \sqrt{1845} \approx 42.95 \)? No, this is confusing. Wait, maybe the problem has simpler values. Let's re-express with correct tangent lengths:

Tangents from a vertex to incircle are equal:

- From \( X \): \( XN = XM = 18 \) (given \( NX = 18 \), \( XM = 18 \))

- From \( Z \): \( ZN = ZL \). \( ZX = 38 \), so \( ZN = ZX - XN = 38 - 18 = 20 \), so \( ZL = 20 \)

- From \( Y \): \( YM = YL \). \( XY = 57 \), so \( YM = XY - XM = 57 - 18 = 39 \), so \( YL = 39 \)

Incenter distances: \( CN = CL = CM \) (inradius). Let's assume \( CN = 18 \) (since \( XN = 18 \), maybe \( CN = 18 \))

Then:

- \( ZL = 20 \) (tangent from \( Z \))

- \( CZ \): In \( \triangle CZN \), \( ZN = 20 \), \( CN = 18 \), so \( CZ = \sqrt{20^2 + 18^2} = \sqrt{400 + 324} = \sqrt{724} \approx 26.9 \) (but this is messy, maybe I made a mistake)

Wait, maybe the given \( CN \) is 18, so \( CN = CL = CM = 18 \)

- \( LY = YL = YM = 57 - 18 = 39 \) (since \( YM = XY - XM = 57 - 18 = 39 \))

- \( CY \): In \( \triangle CYM \), \( YM = 39 \), \( CM = 18 \), so \( CY = \sqrt{39^2 + 18^2} = \sqrt{1521 + 324} = \sqrt{1845} = 9\sqrt{23} \approx 42.95 \)

- \( CM = 18 \) (equal to \( CN \))

- \( CN = 18 \)

- \( CZ \): \( \sqrt{ZN^2 + CN^2} = \sqrt{20^2 + 18^2} = \sqrt{724} = 2\sqrt{181} \approx 26.9 \)

But maybe the diagram has \( ZN = 19 \), so \( ZN = 19 \), \( XN = 18 \), \( ZX = 37 \), but the diagram says 38. This is confusing. Alternatively, maybe the problem has \( ZL = 19 \) (since the diagram has 19 near \( ZN \)), \( CN = 18 \), \( LY = 39 \) (57 - 18), \( CZ = \sqrt{19^2 + 18^2} = \sqrt{361 + 324} = \sqrt{685} \approx 26.17 \), \( CY = \sqrt{39^2 + 18^2} = \sqrt{1521 + 324} = \sqrt{1845} \approx 42.95 \), \( CM = 18 \), \( CN = 18 \).

Given the confusion, but let's proceed with tangent lengths:

1. \( ZL \): Tangent from \( Z \) to incircle, so \( ZL = ZN \). If \( ZN = 38 - 18 = 20 \), then \( ZL = 20 \).

2. \( CZ \): Distance from \( Z \) to \( C \), using Pythagoras: \( CZ = \sqrt{ZN^2 + CN^2} \). If \( CN = 18 \), \( ZN = 20 \), then \( CZ = \sqrt{400 + 324} = \sqrt{724} \approx 26.9 \) (but maybe integer values, so perhaps \( ZN = 19 \), \( CN = 18 \), \( CZ = \sqrt{19^2 + 18^2} = \sqrt{685} \approx 26.17 \))

3. \( LY \): Tangent from \( Y \) to incircle, \( LY = YM = 57 - 18 = 39 \)

4. \( CN = 18 \) (equal to \( CM \))

5. \( CY = \sqrt{LY^2 + CL^2} = \sqrt{39^2 + 18^2} = \sqrt{1845} \approx 42.95 \)

6. \( CM = 18 \) (equal to \( CN \))

But maybe the problem has \( ZL = 19 \) (since the diagram has 19 near \( ZN \)), \( CN = 18 \), \( LY = 39 \), \( CZ = \sqrt{19^2 + 18^2} = \sqrt{685} \), \( CY = \sqrt{39^2 + 18^2} = \sqrt{1845} \), \( CM = 18 \).

Given the initial problem, let's summarize the answers (assuming \( CN = 18 \), \( ZN = 20 \), \( YM = 39 \)):

- \( ZL = 20 \)

- \( CZ = \sqrt{20^2 + 18^2} = \sqrt{724} \) (or 26.9)

- \( LY = 39 \)

- \( CN = 18 \)

- \( CY = \sqrt{39^2 + 18^2} = \sqrt{1845} \) (or 42.95)

- \( CM = 18 \)

But maybe the problem expects integer values, so perhaps the diagram's 19 is correct, \( ZN = 19 \), \( XN = 18 \), \( ZX = 37 \) (but diagram says 38), so maybe a typo. Alternatively, the answer is:

\( ZL = 19 \) (since the diagram has 19), \( CN = 18 \), \( LY = 57 - 18 = 39 \), \( CZ = \sqrt{19^2 + 18^2} = \sqrt{685} \), \( CY = \sqrt{39^2 + 18^2} = \sqrt{1845} \), \( CM = 18 \).

Given the ambiguity, but following tangent lengths:

Answer:

\( ZL = 20 \), \( CZ = \sqrt{724} \), \( LY = 39 \), \( CN = 18 \), \( CY = \sqrt{1845} \), \( CM = 18 \)

(Note: The answer may vary based on diagram interpretation, but this follows tangent segment properties and incenter distance formulas.)