1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
pickupchik [31]
3 years ago
9

In triangle $ABC$, let angle bisectors $BD$ and $CE$ intersect at $I$. The line through $I$ parallel to $BC$ intersects $AB$ and

$AC$ at $M$ and $N$, respectively. If $AB = 17$, $AC = 24$, and $BC = 33$, then find the perimeter of triangle $AMN$.
Thanks!
Mathematics
1 answer:
Umnica [9.8K]3 years ago
8 0

Answer:

41

Step-by-step explanation:

If you work through a series of obscure calculations involving area and the radius of the incircle, they boil down to a simple fact:

... For MN║BC, perimeter ΔAMN = perimeter ΔABC - BC = AB+AC

.. = 17+24 = 41

_____

Wow! Thank you for an interesting question with a not-so-obvious answer.

_____

<em>A little more detail</em>

The point I that you have defined is the incenter—the center of an inscribed circle in the triangle. Its radius is the distance from I to any side, such as BC, for example.

If we use "Δ" to represent the area of the triangle and "s" to represent the semi-perimeter, (AB+BC+AC)/2, then the incircle has radius Δ/s. The area Δ can be computed from Heron's formula by ...

... Δ = √(s(s-a)(s-b)(s-c)) . . . . where a, b, c are the side lengths

For this triangle, the area is Δ = √38480 ≈ 196.1632 units². That turns out to be irrelevant.

The altitude to BC will be 2Δ/(BC), so the altitude of ΔAMN = (2Δ/(BC) -Δ/s). Dividing this by the altitude to BC gives the ratio of the perimeter of ΔAMN to the perimeter of ΔABC, which is 2s.

Putting these ratios and perimeters together, we get ...

... perimeter ΔAMN = (2Δ/(BC) -Δ/s)/(2Δ/(BC)) × 2s

... = (2/(BC) -1/s) × BC × s = 2s -BC

... perimeter ΔAMN = AB +AC

You might be interested in
Hot dogs at a carnival cost 2.99 each plus 7%tax.what is the total cost for one hot dog?
Mars2501 [29]
$3.20
2.99 • 7% = 0.2093
2.99 + 0.2093 = 3.20
3 0
3 years ago
HELP PLEASEEEE. The question is attached. I would appreciate if someone could help me.
Blizzard [7]

a counterclockwise rotation about the origin of 90°

The coordinates of P(3, 3), Q(5, 3), R(5, 7)

The coordinates of P'(- 3, 3 ), Q'(- 3, 5), R'(- 7, 5)

Note that the y-coordinate of the image is the negative of the original, while the x-coordinate of the original becomes the y-coordinate of the image

The rotation which does this is a counterclockwise rotation about the origin of 90°

a point (x, y ) → (- y, x )


8 0
3 years ago
Javier monthly net income is $1,500. He requests that his bank deposit 1/5 of his monthly earnings into his savings account. how
Harman [31]
1500 * 1/5
1500 * 0.2 = 300
300 * 12 = 3600
Javier contributes $3,600 a year to his savings.
5 0
3 years ago
Please Help Me I really need Accurate answer Can anyone help me?​
DanielleElmas [232]

Answer:

mathematics make our brain fast and sharp

6 0
3 years ago
Kelly buys a shirt for $14 dollars and pays the store clerk $20 how much change will kelly get back
Ilia_Sergeevich [38]

Answer:

6

Step-by-step explanation:

6 maybe

8 0
3 years ago
Other questions:
  • Ayuda por favor no entiendo esta pregunta?????
    14·1 answer
  • Please help quick! Write the quadratic function f(x) = 3(x + 1)(x – 7) in standard form. (Answers attached below)
    8·1 answer
  • Write 6^8 • 6^7 as a single power
    7·1 answer
  • 5x+15y=75 in standard form
    5·2 answers
  • PLEASE HELP!!!!!!!!!!
    14·1 answer
  • Don't mind the writings in pen, i didn't know what i was doing... PLS HELP ASAP​
    10·1 answer
  • ASK YOUR TEACHER
    13·1 answer
  • Please help me with this!!
    11·1 answer
  • Brainliest if you help me ….
    7·1 answer
  • Rachel is mailing packages. Each small package costs her $2.60 to send. Each large package costs her$4.10. How much will it cost
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!