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frika

Answer:

16 un.

Step-by-step explanation:

In right triangle ABC:

  • m∠C = 90°;
  • m∠BAC = 2m∠ABC;
  • BC = 24;
  • AL is a bisector of angle A.

The sum of the measures of all interior angles in triangle  is always 180°, then

[tex]m\angle ABC+m\angle BCA+m\angle BAC=180^{\circ},\\ \\m\angle ABC+90^{\circ}+2m\angle ABC=180^{\circ},\\ \\3m\angle ABC=180^{\circ}-90^{\circ}=90^{\circ},\\ \\m\angle ABC=30^{\circ},\\ \\m\angle BAC=60^{\circ}.[/tex]

In right triangle the leg that is opposite to tha angle 30° is half of the hypotenuse. This means that

[tex]AC=\dfrac{AB}{2},\\ \\AB=2AC.[/tex]

By the Pythagorean theorem,

[tex]AB^2=AC^2+BC^2,\\ \\(2AC)^2=24^2+AC^2,\\ \\4AC^2=576+AC^2,\\ \\3AC^2=576,\\ \\AC^2=192,\\ \\AC=8\sqrt{3}\ un.,\\ \\AB=2AC=16\sqrt{3}\ un.[/tex]

Let AL be the angle A bisector. By bisector property,

[tex]\dfrac{AC}{AB}=\dfrac{CL}{LB},\\ \\\dfrac{8\sqrt{3}}{16\sqrt{3}}=\dfrac{CL}{24-CL},\\ \\\dfrac{1}{2}=\dfrac{CL}{24-CL},\\ \\24-CL=2CL,\\ \\3CL=24,\\ \\CL=8\ un.[/tex]

Use the Pythagorean theorem for the right triangle ACL:

[tex]AL^2=AC^2+CL^2,\\ \\AL^2=(8\sqrt{3})^2+8^2,\\ \\AL^2=192+64=256,\\ \\AL=16\ un.[/tex]

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MrRoyal

The bisector of a line segment divides the segment into bits.

The value of AL is 16

The given parameters are:

[tex]\angle C = 90^o[/tex]

[tex]\angle BAC = 2\angle ABC[/tex]

[tex]BC =24[/tex]

The angles in a triangle add up to 180 degrees.

So, we have:

[tex]\angle C + \angle BAC + \angle ABC = 180[/tex]

This gives

[tex]90 + 2\angle ABC + \angle ABC = 180[/tex]

[tex]90 + 3\angle ABC = 180[/tex]

Subtract 90 from both sides

[tex]3\angle ABC = 90[/tex]

Divide both sides by 3

[tex]\angle ABC = 30[/tex]

Substitute [tex]\angle ABC = 30[/tex]  in [tex]\angle BAC = 2\angle ABC[/tex]

[tex]\angle BAC = 2\times 30[/tex]

[tex]\angle BAC = 60[/tex]

So, we have:

[tex]\angle BAC = 60[/tex]

[tex]\angle ABC = 30[/tex]

[tex]\angle C = 90^o[/tex]

The ratio of the sides of a 30-60-90 triangle is:

[tex]AC:BC:AB=1:\sqrt 3:2[/tex]

Given that BC = 24, the ratio becomes

[tex]AC:24:AB=1:\sqrt 3:2[/tex]

Split

[tex]AC:24=1:\sqrt 3[/tex]

[tex]24:AB=\sqrt 3:2[/tex]

Express [tex]AC:24=1:\sqrt 3[/tex] as fraction

[tex]\frac{AC}{24} = \frac{1}{\sqrt 3}[/tex]

Multiply both sides by 24

[tex]AC = \frac{24}{\sqrt 3}[/tex]

Rationalize

[tex]AC = \frac{24\sqrt 3}{3}[/tex]

[tex]AC = 8\sqrt 3[/tex]

Length AL is then calculated as:

[tex]AL = \frac{2}{\sqrt 3}\times AC[/tex]

This gives

[tex]AL = \frac{2}{\sqrt 3}\times 8\sqrt 3[/tex]

[tex]AL = 2\times 8[/tex]

[tex]AL = 16[/tex]

Hence, the value of AL is 16

Read more about bisectors at:

https://brainly.com/question/7698080

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