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Ganezh [65]
3 years ago
12

3)A triangle has all integer side lengths and two of those sides have lengths 9 and 16. Consider the altitudes to the three side

s. What is the largest possible value of the ratio of any two of those altitudes

Mathematics
2 answers:
Bad White [126]3 years ago
6 0

Answer:

2 is the largest possible value.

Step-by-step explanation:

Given Data:

Length of one side       = 9

Length of second side = 16

Let AB and BC sides of triangle be of length 9 and 16 respectively as shown in figure attached.

Now, let sides AD and CF be the respective altitudes.

Also, ∠ABC = ∅ (as shown in figure)                                    

If AD and CF are the respective altitudes then,

we have

       AD = 9Sin∅ ;

       CF = 16Sin∅;

By dividing both sides, we get

       AD/CF = 9/16

This equation shows that is independant of angle ∅.

Now, let ∠BAC = α

Now we have, BE = 9 sinα

and                   FC = AC sinα

By dividing both sides, we get

       BE/FC=9/AC

Similarly we also have,

       BE/AD = 16/AC

ABC is a triangle as long as length of AC is ithin range of 8 to 24 i.e, 8≤AC≤24 (because sum of any two sides of triangle should be greater than length of third side)

Using these values we get ranges of:

9/24 ≤ BF/FC ≤ 9/8  ;  2/3 ≤ BE/AD ≤ 2

So,

2 is the largest possible value of the ratio of any two of these altitudes.

Art [367]3 years ago
3 0

Answer: 2

Step-by-step explanation:

16/9=1.7

=Approximately=2

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The angle bisectors of AXYZ are XG, YG, and ZG. They meet at a single point G.
Harrizon [31]

Answer:

m∠FXD = 36°

EG = 10

m∠FZG = 24°

Step-by-step explanation:

In triangle XYZ, G is the incenter of the triangle.

Since, m∠FXG = 18°

And m∠FXD = 2(m∠FXG)

                     = 2 × 18°

                     = 36°

Since, point G is equidistant from all sides (Property of incenter of a triangle)

Therefore, DG = EG = GF = 10

Since, m∠X + m∠Y + m∠Z = 180°

2(m∠FXG) + m∠DYE + 2(m∠FZG) = 180°

2(18)° + 96° + 2(m∠FZG) = 180°

2(m∠FZG) = 180° - 132°

m∠FZG = 24°

3 0
3 years ago
pick any number and add 1 to it. find the sum of the new number and the original number. Add 9 to the sum. Divide the new sum by
Monica [59]
The correct answer is x+1<span> </span>
7 0
3 years ago
I need help asap!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
almond37 [142]

Answer:

Answers

Step-by-step explanation:

1.) 4 weeks = 28 days

12 x 4 = 48 in

2.) 13.95 ÷ 4 = 4.65

3.) 9 ÷ 4 = $2.25 per pound

4.) yes, 4 x 4 = 16, 9 x 4 = 36

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7 0
2 years ago
2.The diagonals of a rhombus are in the ratio 5:12. If its perimeter is 104 CM, findthe lengths of the sides and the diagonals.​
coldgirl [10]

Answer:

Lenghts of the sides: 26\ cm

Lenghts of the diagonals: 48\ cm and 20\ cm

Step-by-step explanation:

Look at the rhombus ABCD shown attached, where AC and BD de diagonals of the rhombus.

The sides of a rhombus have equal lenght. Then, since the perimeter of this one is 104 centimeters, you can find the lenght of each side as following:

AB=BC=CD=DA=\frac{104\ cm}{4}= 26\ cm

You know that the diagonals are in the ratio 5:12

Then, let the diagonal AC be:

AC=12x

This means that AE is:

AE=\frac{12x}{2}=6x

And let the diagonal BD be:

BD=5x

So BE is:

BE=\frac{5x}{2}=2.5x

Since the diagonals of a rhombus are perpendicular to each other, four right triangles are formed, so you can use the Pythagorean Theorem:

a^2=b^2+c^2

Where "a" is the hypotenuse and "b" and "c" are the legs.

In this case, you can choose the triangle ABE. Then:

a=AB=26\\b=AE=6x\\c=BE=2.5x

Substituting values and solving for "x", you get:

26^2=(6x)^2+(2.5x)^2\\\\676=36x^2+6.25x^2\\\\\sqrt{\frac{676}{42.25}}=x\\\\x=4

Therefore, the lenghts of the diagonals are:

AC=12(4)\ cm=48\ cm

BD=5(4)\ cm=20\ cm

8 0
3 years ago
What is 4/12 + 1/12 + 3/12
vesna_86 [32]

Answer:

8/12

Step-by-step explanation:

6 0
3 years ago
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