Construct two altitudes of the triangle, one fromto a pointon, and the one stated in the question.

is isosceles, so the mediancuts 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 altitudeby 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.

Sinceis, by definition, perpendicular to, it divides the triangle into 45-45-90 triangleand the 30-60-90 triangle.

Letbe the length of. By the 45-45-90 Theorem,, and, the legs of, are congruent, so; by the 30-60-90 Theorem, long legofhas lengthtimes that of, or. Therefore, the length ofis:

We are given that, so

and

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

The areaof a triangle is half the product of the lengths of a base and that of its corresponding altitude. If we letand(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 altitudesand; note thathas been extended to a rayto facilitate the location of the point.

For the sake of simplicity, we will call the measure of1; the ratio is the same regarless of the actual measure, and the measure ofwilll give us the desired ratio.

Since, and, by definition, is perpendicular to,is a 30-60-90 triangle. By the 30-60-90 Theorem, hypotenuseofhas 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, makinga 30-60-90 triangle. By the 30-60-90 Theorem, shorter legofhas half the length of hypotenuse, so; also, longer leghas lengthtimes this, or.

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

is shown below, along with altitude; note thathas been extended to a rayto 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, makinga right triangle anda 30-60-90 triangle. By the 30-60-90 Triangle Theorem, shorter legofhas half the length of hypotenuse—that is, half of 48, or 24; longer leghas lengthtimes 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 ofis three fourths that of that of.