Mathematics, 26.06.2019 05:00 fernandezvela27

An angle bisector ac divides a trapezoid abcd into two similar triangles △abc and △acd. find the perimeter of this trapezoid if the leg ab=9 cm and the leg cd=12 cm.

Answers: 1

Show answers

Answers

Answer from: hjamya17

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,

$\frac{AB}{CD} =\frac{BC}{AC} =\frac{AC}{AD}$

But, $\frac{AB}{CD} =\frac{9}{12} =\frac{3}{4}$

Therefore, $\frac{BC}{AC} =\frac{3}{4}$

$\frac{9}{AC} =\frac{3}{4}$

$AC=(\frac{4}{3} )(9)$

AC = 12 cm

Also,

$\frac{AC}{AD} =\frac{3}{4}$

$\frac{12}{AD} =\frac{3}{4}$

$AD=(\frac{4}{3} )(12)$

AD = 16 cm

Now, the perimeter of the trapezoid = AB + BC + CD + AD

= 9 + 9 + 12 + 16

= 46 cm

An angle bisector ac divides a trapezoid abcd into two similar triangles △abc and △acd. find the per

Answer from: krisbarnett5353

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:

$\rm \angle DAC = \angle CAB$ ----- (1)

$\rm \angle DAC = \angle ACB$ (alternate interior angles) ---- (2)

From (1) and (2): $\rm \angle CAB = \angle ACB$

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.

$\rm \dfrac{AB}{CD}=\dfrac{BC}{AC}=\dfrac{AC}{AD}$

$\rm \dfrac{AB }{CD}=\dfrac{9}{12}=\dfrac{3}{4}$

Hence, $\rm \dfrac{3}{4}=\dfrac{BC}{AC}$

$\rm \dfrac{9}{AC}=\dfrac{3}{4}$

AC = 12 cm

$\rm \dfrac{AC}{AD}=\dfrac{3}{4}$

$\rm \dfrac{12}{AD}=\dfrac{3}{4}$

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:

link

Answer from: Quest

q1 a person or thing situated away or detached from the main body or system. a cluster is a a mash of many things.

q2 do

step-by-step explanation:

Answer from: Quest

true

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 17:30

Lindsay used two points, (x,y; ) and (+2.82), to find the equation of the line, y = mx + b, that passes through the points. y2-y, first, she used the definition of slope and determined that the value of mis x, - . given this information, which expression must represent the value of b?

Answers: 2

Answer

Mathematics, 21.06.2019 18:30

Afarm that has 4 x 10^2 acres has 1.2 x 10^5 cattle. what is the average number of cattle per acre? write the answer in scientific notation. answere asap

Answers: 2

Answer

Mathematics, 21.06.2019 19:10

Find the roots of the polynomial function f(x) = x^3 + 2x^2 + x

Answers: 2

Answer

Mathematics, 21.06.2019 22:00

Find the greatest common factor of the followig monomials 46g^2h and 34g^6h^6

Answers: 1

Answer

You know the right answer?

An angle bisector ac divides a trapezoid abcd into two similar triangles △abc and △acd. find the per...