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In right triangle ΔABC (m∠C = 90°), point P is the intersection of the angle bisectors of the acute angles. The distance from P to the hypotenuse is equal to 4 in. Find the perimeter of △ABC if AB = 20 in.

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sqdancefan
The intersection point of the angle bisectors is the incenter of the triangle, the center of the incircle, which has radius 4.

One of the properties of the incircle is that the distances (d) from vertex C to the nearest touchpoints are equal and have the value
  d = 1/2(a +b -c)
where a, b, c are the lengths of the sides opposite angles A, B, C, respectively.

We know the distance from the right-angle vertex C to the incircle touchpoints is the radius of that circle, 4 in. And, we know the length of the hypotenuse (c) is 20 in. This lets us find the sum a+b as ...
  4 = (1/2)(a+b -20) . . .. fill known information into the formula
  8 = (a+b -20) . . . . . . . multiply by 2
  28 = a+b . . . . . . . . . .. the total length of the two legs of the triangle

The perimeter is the sum of the leg lengths and the hypotenuse length.
  perimeter = 28 in + 20 in = 48 in

The perimeter of ΔABC is 48 in.


_____
It turns out that the triangle is a 3-4-5 triangle scaled by a factor of 4, so having sides 12, 16, and 20.

Perimeter of ΔABC is equals to 48 inches.

What is Perimeter?

" Perimeter of any two dimensional figure is the total length around the shape of the given figure."

Formula used

Pythagoras theorem

(Hypotenuse)² = (Adjacent side)² + ( Opposite side)²

(a ± b)²  = a² ± 2ab + b²

According to the question,

In ΔABC ,

∠C=90°

'P' is the intersection of the angle bisectors of the acute angles.

Distance from P to the hypotenuse = 4inches

As per diagram

By theorem : Angle bisector of a triangle are concurrent.

'Q' , 'R' and 'S' are the tangent points of incircle drawn through center 'P'.

Let 'x' be the length BR

BR= BQ        (tangents drawn from a exterior point have equal length)

Therefore, AQ = 20-x                           (AB = 20inches)

AQ= AS

In quadrilateral PRCS,

PR = PS              (inradius)

CR =CS               (tangent length)

All the interior angles are of measure 90°.

Therefore, quadrilateral PRCS is a square.

CR = CS= PR=PS = 4inches

Therefore,

BC = x + 4

AC = 20 -x +4

     = 24 -x

By Pythagoras theorem,

[tex]AB^{2} =AC^{2} +BC^{2}[/tex]

Substitute the value of AC and BC we have,

(20)² = (x+4)² +(24 -x)²

⇒400 = x² + 8x + 16 + 576 - 48x +x²

⇒2x² - 40x + 192 =0

⇒ x² -20x +96 =0

⇒ (x- 12) (x -8)=0

⇒ x = 12 or x =8

When x =12

AC = 12 , BC = 16

or x=8

⇒ AC = 16 , BC = 12

Therefore , Perimeter of a ΔABC = AB +BC + AC

                                                       = 20 + 12 + 16

                                                       = 48 inches

Hence, Perimeter of ΔABC is equals to 48 inches.

Learn more about Perimeter here

https://brainly.com/question/27566657

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