The proportions of corresponding sides of similar triangles must be equal. Therefore, . .

To solve this equation, we need to calculate the length of the height with the Pythagorean Theorem.

We could also recognize that since  and , the triangle is a Pythagorean Triple, in other words, its sides will be in ratio  where  is a constant.

Here  and therefore, the length of height BD must be , which is our final answer.

If the largest angle of the obtuse isosceles triangle is  , then this is the unique angle in between the two sides with an equal length of .  We can imagine that the height of this isosceles triangle is simply the third side of a triangle formed by half of its base and the length of either equal side. That is, if we bisected the    angle with a line perpendicular to the base of the obtuse isosceles triangle, this line would be the height of the triangle. If we bisected the    angle, we would have two congruent triangles with angles of   between the height and each side of equal length. This means the cosine of that angle will be equal to the length of the height over the length of either equal side, which gives us:

If one measure of an obtuse isosceles triangle is  ,  then this is obviously the unique angle that classifies the triangle as obtuse, which tells us that this is the angle between the two sides with an equivalent length of  .  The height of the triangle is given by a line that bisects this angle.  This tells us that the angle between the height and the sides of equivalent length is  ,  and because we know the length of the equivalent sides we can solve for the height as follows, where    is the height of the triangle and    is the length of the equivalent sides:

 is shown below, along with altitude .

By the Isosceles Triangle Theorem, since ,  is isosceles with . By the Hypotenuse-Leg Theorem, the altitude cuts  into congruent triangles  and , so ; this makes  the midpoint of .  has length 42, so  measures half this, or 21.

Also, 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 length equal to that of longer leg  divided by ; that is, 

 is shown below, along with altitude .

Since  is, by definition, perpendicular to , it divides the triangle into 45-45-90 triangle  and 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, short leg  of  has as its length that of  divided by , or . Therefore, the length of  is:

We are given that , so 

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

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 .