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
8090 [49]
2 years ago
12

100 POINTS

Mathematics
2 answers:
Anvisha [2.4K]2 years ago
7 0

Answer:

6√3 ± 3

Step-by-step explanation:

AC is the radius of Circle C ⇒ radius = 12

DB is the radius of Circle D ⇒ radius = 9

Therefore, CD = AC + DB = 12 + 9 = 21

AE is the radius of Circle E ⇒ radius = 3

BF is the radius of Circle F ⇒ radius = 3

If AB is the common tangent of Circles C, D, E and F, then:

∠CAB =∠AEF = ∠DBA = ∠BFE = 90°

(as the tangent to a circle is always <u>perpendicular</u> its the radius).

Also AB = EF

To find the possible radii of a circle whose center is the midpoint of EF and is tangent to both circles E and F, we first need to find the length of EF (marked by x on the attached diagram).

CD is the hypotenuse of a right triangle with base equal to EF and a smaller leg of the difference between the radii of Circles C and D.

(Refer to the green shaded triangle on attachment 1)

Therefore, to find EF (x), use Pythagoras' Theorem a² + b² = c²
(where a and b are the legs, and c is the hypotenuse, of a right triangle)

⇒ a² + b² = c²

⇒ 3² + x² = CD²

⇒ 3² + x² = (12 + 9)²

⇒ 9 + x² = 441

⇒ x² = 432

⇒ x = √432

⇒ x = 12√3

⇒ EF = 12√3

As the radius of Circles E and F is 3, the possible diameter of the circle with its center as the midpoint of EF is 6 less or 6 more than the length of EF.

⇒ diameter = EF ± 6

                   = 12√3 ± 6

As the radius is <u>half of the diameter</u>, the possible radii of a circle whose center is the midpoint of EF and is tangent to both circles E and F is:

⇒ radius = diameter ÷ 2

               = (12√3 ± 6) ÷ 2

               = 6√3 ± 3

(see attachments 2 and 3 → the possible circles are shown in orange)

Shkiper50 [21]2 years ago
5 0

Answer:

  6√3 ±3 ≈ {7.392, 13.392}

Step-by-step explanation:

The length of AB is the long side of a right triangle with hypotenuse CD and short side (AC -BD). The desired radius values will be half the length of EF, with AE added or subtracted.

__

<h3>length of AB</h3>

Radii AC and BD are perpendicular to the points of tangency at A and B. They differ in length by AC -BD = 12 -9 = 3 units.

A right triangle can be drawn as in the attached figure, where it is shaded and labeled with vertices A, B, C. Its long leg (AB in the attachment) is the long leg of the right triangle with hypotenuse 21 and short leg 3. The length of that leg is found from the Pythagorean theorem to be ...

  AB = √(21² -3²) = √432 = 12√3

<h3>tangent circle radii</h3>

This is the same as the distance EF. Half this length, 6√3, is the distance from the midpoint of EF to E or F. The radii of the tangent circles to circles E and F will be (EF/2 ±3). Those values are ...

  6√3 ±3 ≈ {7.392, 13.392}

You might be interested in
A right triangle has a leg 6 and hypotenuse of 19. Find the missing length?
Ne4ueva [31]

Answer:

18

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Increase 58.50 by 10% working
natka813 [3]

Answer:

Final Value = 58.50 × (1 + 10/100)

Final Value = 58.50 × (1 + 0.1)

Final Value = 58.50 × (1.1)

Final Value = 64.35

8 0
3 years ago
Read 2 more answers
How can I solve this proof?? PLZ HELP I DONT UNDERSTAND!! Thanks !!
love history [14]

Hello from MrBillDoesMath!

Answer:

See below


Discussion:

Triangles BAD and BCD are congruent by the SAS postulate.

          * Side AD = DC because BD is the perpendicular bisector  of AC.

          * Angle ADB and CDB are both right angles, and hence equal, because

            BD is the perpendicular bisector of AC)


Corresponding parts of congruent triangles are equal so angle BAD = angle BCD,  which was to be proven.


Regards,  

MrB

8 0
3 years ago
How would the graph of the relation y=3x-2 change if the 3 and -2 were both doubled? 
Lena [83]
<u>y = 3x - 2</u>

-- The ' 3 ' is the slope. If you double it to ' 6 ', the line is steeper.

-- The ' -2 ' is the y-intercept. If you double it to ' -4 ', the line crosses the y-axis lower.

Choice <u>(a)</u> says that . . . "... steeper, and lower y-intercept".

6 0
3 years ago
What is the true solution to the logarithmic equation below?
Oliga [24]

Answer:

  • x = 2

Step-by-step explanation:

<u>Given:</u>

  • log ₄ (2x) = 1

<u>Solving for x:</u>

  • 2x = 4¹
  • 2x = 4
  • x = 2
3 0
2 years ago
Other questions:
  • How could you use the conversion table on page 618 to convert 700 cenimeters to meters
    5·1 answer
  • 5L/s=__________kl/min
    12·1 answer
  • If Rebekah had 28 Jolly Ranchers, ate 3, gave 5 to her friend Hayley, and 5 to her friend Audrey, how many Jolly Ranchers does s
    8·2 answers
  • on a floor plan, a room measures 2 inches by 5 inches. if the scale is 1 inch = 4 feet, what is the actual size of the room?
    14·1 answer
  • What is the cost of a $1200 washing machine that was on sale for a 1/5 discount?
    13·2 answers
  • A pillow that usually costs $18 is marked down 25%. Which expressions can be used to find the new price of the pillow? Check all
    15·2 answers
  • Help someone please!
    13·1 answer
  • Question: Tell whether each function is linear or non-linear.
    13·1 answer
  • A baker needs to have 15.5 pounds of flour on hand for the weekend bread so he looks in the cupboard and finds a bag that has 4.
    9·1 answer
  • Nearest ten thousand 384400
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!