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

We could also recognize that sinceand,三角形是毕达哥拉斯三words, its sides will be in ratiowhereis a constant.

Hereand 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 theangle with a line perpendicular to the base of the obtuse isosceles triangle, this line would be the height of the triangle. If we bisected theangle, we would have two congruent triangles with angles of高度和每一方之间的长度相等。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,那么这显然是cla的独特角度ssifies 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, whereis the height of the triangle andis 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 cutsinto congruent trianglesand, so;this makesthe midpoint of.has length 42, someasures half this, or 21.

一个lso, 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 legdivided by;that is,

is shown below, along with altitude.

Sinceis, by definition, perpendicular to, it divides the triangle into 45-45-90 triangleand 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, short legofhas as its length that ofdivided by, or. Therefore, the length ofis:

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 fromto a pointon一个规定的问题。

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;此外,长腿has 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.