Question

3)A triangle has all integer side lengths and two of those sides have lengths 9 and 16. Consider the altitudes to the three sides. What is the largest possible value of the ratio of any two of those altitudes

Answer 1

2 is the largest possible value.

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.

Answer 2

Answer: 2

Explanation:

16/9=1.7

=Approximately=2