Construct two altitudes of the triangle, one from  to a point  on , and the one stated in the question. 

 is isosceles, so the median  cuts it into two congruent triangles;  is the midpoint, so (as marked above)  has length half that of , or half of 10, which is 5. By the Pythagorean Theorem,

The area of a triangle is one half the product of the length of any base and its corresponding height; this is , but it is also . Since we know all three sidelengths other than that of , we can find the length of the altitude by setting the two expressions equal to each other and solving for :

To find out what two integers this falls between, square it:

Since , it follows that .

 is shown below, along with altitude .

Since  is, by definition, perpendicular to , it divides the triangle into 45-45-90 triangle  and the 30-60-90 triangle .

Let  be the length of . By the 45-45-90 Theorem, , and , the legs of , are congruent, so ; by the 30-60-90 Theorem, long leg  of  has length  times that of , or . Therefore, the length of  is:

We are given that , so 

and 

We can simplify this by multiplying both numerator and denominator by , thereby rationalizing the denominator:

The area  of a triangle is half the product of the lengths of a base and that of its corresponding altitude. If we let  and  (height) stand for those lengths, respectively, the formula is

,

which can be restated as:

It follows that in the same triangle, the length of an altitude is inversely proportional to the length of the corresponding base, so the longest base will correspond to the shortest altitude, and vice versa.

Since, in descending order by length, the sides of the triangle are

,

their corresponding altitudes are, in ascending order by length,

.

 is shown below, along with altitudes  and ; note that  has been extended to a ray  to facilitate the location of the point . 

For the sake of simplicity, we will call the measure of  1; the ratio is the same regarless of the actual measure, and the measure of  willl give us the desired ratio. 

Since , and , by definition, is perpendicular to ,  is a 30-60-90 triangle. By the 30-60-90 Theorem, hypotenuse  of  has length twice that of short leg , so .

Since an exterior angle of a triangle has as its measure the sum of those of its remote interior angles, 

.

By defintiion of an altitude,  is perpendicular to , making  a 30-60-90 triangle. By the 30-60-90 Theorem, shorter leg  of  has half the length of hypotenuse , so ; also, longer leg  has length  times this, or .

The correct choice is therefore that the ratio of the lengths is .

 is shown below, along with altitude ; note that  has been extended to a ray  to facilitate the location of the point . 

Since an exterior angle of a triangle has as its measure the sum of those of its remote interior angles, 

By definition of an altitude,  is perpendicular to , making  a right triangle and  a 30-60-90 triangle. By the 30-60-90 Triangle Theorem, shorter leg  of  has half the length of hypotenuse —that is, half of 48, or 24; longer leg  has length  times this, or , which is the correct choice.

 is shown below, along with altitude .

Since , and , by definition, is perpendicular to ,  is a 30-60-90 triangle. By the 30-60-90 Triangle Theorem, , as the shorter leg of , has half the length of hypotenuse ; this is half of 30, or 15.

The area of a triangle is one half the product of the length of any base and its corresponding height; this is , but it is also . Set these equal, and note the following:

That is, the length of  is three fourths that of that of .