Respuesta :
Answer:
46 cm
Step-by-step explanation:
It is given that the angle bisector AC divides the trapezoid ABCD into two similar triangles Δ ABC and Δ ACD.
Let us first find the corresponding sides of triangles Δ ABC and Δ ACD.
Since AC is the angle bisector of ∠A,
∠DAC = ∠CAB --- (1)
Also, since ∠DAC and ∠ACB are alternate interior angles,
∠DAC = ∠ACB --- (2)
From (1) and (2),
∠CAB = ∠ACB
Therefore, in Δ ABC,
BC = AB (sides opposite to equal angles are equal)
So, BC = 9 cm.
Now, since ∠DAC = ∠ACB and ∠DAC = ∠CAB, we can take either ∠ACB or ∠CAB as the angle corresponding to ∠DAC. Let us take ∠ACB as the corresponding angle of ∠DAC.
So, the side opposite to ∠DAC in Δ ACD is the corresponding side to the side opposite to ∠ACB in Δ ABC.
That is, the side CD in Δ ACD is the corresponding side to the side AB in Δ ABC.
Now, suppose if the side BC in Δ ABC corresponds to side AD of Δ ACD, then the remaining side AC of Δ ABC should correspond to side AC of Δ ACD which is not possible since they are congruent.
So, the side BC in Δ ABC should correspond to side AC of Δ ACD and the remaining side AC of Δ ABC should correspond to side AD of Δ ACD.
Therefore, we have,
[tex]\frac{AB}{CD} =\frac{BC}{AC} =\frac{AC}{AD}[/tex]
But, [tex]\frac{AB}{CD} =\frac{9}{12} =\frac{3}{4}[/tex]
Therefore, [tex]\frac{BC}{AC} =\frac{3}{4}[/tex]
[tex]\frac{9}{AC} =\frac{3}{4}[/tex]
[tex]AC=(\frac{4}{3} )(9)[/tex]
AC = 12 cm
Also,
[tex]\frac{AC}{AD} =\frac{3}{4}[/tex]
[tex]\frac{12}{AD} =\frac{3}{4}[/tex]
[tex]AD=(\frac{4}{3} )(12)[/tex]
AD = 16 cm
Now, the perimeter of the trapezoid = AB + BC + CD + AD
= 9 + 9 + 12 + 16
= 46 cm
The perimeter of trapezoid ABCD is 46 cm and this can be determined by using the properties of the triangle.
Given :
An angle bisector AC divides a trapezoid ABCD into two similar triangles △ABC and △ACD.
The leg AB=9 cm and the leg CD=12 cm.
Given that AC is the angle bisector of angle A, therefore:
[tex]\rm \angle DAC = \angle CAB[/tex] ----- (1)
[tex]\rm \angle DAC = \angle ACB[/tex] (alternate interior angles) ---- (2)
From (1) and (2): [tex]\rm \angle CAB = \angle ACB[/tex]
SIdes opposite to equal angles are equal, therefore:
BC = AB
Hence, BC = 9 cm
Side BC in triangle ABC should correspond to side AC of triangle ACD and the remaining side AC of triangle ABC should correspond to side AD of triangle ACD.
[tex]\rm \dfrac{AB}{CD}=\dfrac{BC}{AC}=\dfrac{AC}{AD}[/tex]
[tex]\rm \dfrac{AB }{CD}=\dfrac{9}{12}=\dfrac{3}{4}[/tex]
Hence, [tex]\rm \dfrac{3}{4}=\dfrac{BC}{AC}[/tex]
[tex]\rm \dfrac{9}{AC}=\dfrac{3}{4}[/tex]
AC = 12 cm
[tex]\rm \dfrac{AC}{AD}=\dfrac{3}{4}[/tex]
[tex]\rm \dfrac{12}{AD}=\dfrac{3}{4}[/tex]
AD = 16 cm
Perimeter of Trapezoid = AB + BC + CD + AD
= 9 + 9 + 12 + 16
= 46 cm
For more information, refer to the link given below:
https://brainly.com/question/23450266
