Respuesta :

MrRoyal

Answer:

[tex]Perimeter = 38.6[/tex]

Step-by-step explanation:

Given

[tex]A\ B = 13[/tex]

[tex]B\ F = 9[/tex]

[tex]F\ C = 5.6[/tex]

Required

Calculate the perimeter of A B C

To calculate the perimeter of A B C, we sum up the sides of the triangle

i.e.

[tex]Perimeter = A B + B\ C + A\ C[/tex]

AB is given in the question already.

So, we need to calculate B F and F C

To solve for BF, we consider triangle B F C

Using Pythagoras Theorem, we have:

[tex]B\ C^2 = B\ F^2 + F\ C^2[/tex]

Substitute values for BF and FC;

[tex]B\ C^2 = 9^2 + 5.6^2[/tex]

[tex]B\ C^2 = 81 + 31.36[/tex]

[tex]B\ C^2 = 112.36[/tex]

[tex]B\ C^2= \sqrt{112.36[/tex]

[tex]B\ C = 10.6[/tex]

Next, we calculate A C.

To calculate A C, we need to calculate A F, considering triangle B F A

Using Pythagoras Theorem, we have:

[tex]A\ B^2 = B\ F^2 + A\ F^2[/tex]

Substitute values for B F and A B;

[tex]13^2 = 9^2 + A\ F^2[/tex]

[tex]169= 81 + A\ F^2[/tex]

[tex]A\ F^2 = 169 - 81[/tex]

[tex]A\ F^2 = 88[/tex]

[tex]A\ F = \sqrt{88[/tex]

[tex]A\ F = 9.4[/tex]

Since F lies on line A C:

[tex]A\ C = A\ F + F\ C[/tex]

[tex]A\ C = 9.4 + 5.6[/tex]

[tex]A\ C = 15.0[/tex]

Recall that:

[tex]Perimeter = A\ B + B\ C + A\ C[/tex]

[tex]Perimeter = 13 + 10.6 + 15.0[/tex]

[tex]Perimeter = 38.6[/tex]

Hence, the perimeter is 38.6

akposevictor

Given that P is the orthocenter of triangle ABC, applying the properties of an orthocenter of a triangle and the Pythagorean Theorem, each measure and the perimeter of △ABC are:

  • AB = 13
  • BC = 10.6
  • AC = 15
  • Perimeter = 38.6

Recall:

  • An altitude is the line that connects the vertex of a triangle to the opposite side of which it is perpendicular to that side.
  • Orthocenter is the point where all three altitudes intersect each other.

Given that P is the orthocenter of triangle ABC, where,

  • AB = 13
  • BF = 9
  • FC = 5.6

Perimeter of △ABC = AB + BC + AC

First, we need to find BC and AC.

△BFC is a right triangle, apply Pythagorean theorem to find BC:

[tex]BC = \sqrt{BF^2 + FC^2}[/tex]

  • Substitute

[tex]BC = \sqrt{9^2 + 5.6^2}\\\\\mathbf{BC = 10.6}[/tex]

△BFA is a right triangle, apply Pythagorean theorem to find FA:

[tex]FA = \sqrt{AB^2 - BF^2}[/tex]

  • Substitute

[tex]FA = \sqrt{13^2 - 9^2}\\\\\mathbf{FA = 9.4}[/tex]

AC = FA + FC

  • Substitute

AC = 9.4 + 5.6

AC = 15

Perimeter of △ABC = AB + BC + AC

  • Substitute

Perimeter of △ABC = 13 + 10.6 + 15

Perimeter of △ABC = 38.6 units

Therefore, given that P is the orthocenter of triangle ABC, applying the properties of an orthocenter of a triangle and the Pythagorean Theorem, each measure and the perimeter of △ABC are:

  • AB = 13
  • BC = 10.6
  • AC = 15
  • Perimeter = 38.6

Learn more here:

https://brainly.com/question/2515607

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