The three sides of a scalene triangle have different measures. One measure  cannot have is 12, but this is not a choice.

It cannot be true that . Since the perimeter is 

, we can find out what other value can be eliminated as follows:

Therefore, if , then , and the triangle is not scalene. 9 is the correct choice.

We are looking for ways to add three primes to yield a sum of 43. Two or all three (since an equilateral triangle is considered isosceles) must be equal (although, since 43 is not a multiple of three, only two can be equal).

We will set the shared sidelength of the congruent sides to each prime number in turn up to 19:

By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate the first three. , , and  include numbers that are not prime (21, 15, 9). This leaves us with only one possibility:

 - greatest length 19

19 is the correct choice.

The three sides of  are the midsegments of , so  is similar to .

By the Triangle Midsegment Theorem, each is parallel to one side of . 

By the same theorem, each has length exactly half of that side, giving  twice the perimeter of .

But since the sides of  have twice the length of those of , the area of , which varies directly as the square of a sidelength, must be four times that of .

The correct choice is the one that asserts that the area of  is twice that of .

In a triangle, the angle of greatest measure is opposite the side of greatest measure, and the angle of least measure is opposite the side of least measure. , so their opposite angles are ranked in this order - that is, .

Corresponding angles of similar triangles are congruent, so, since , .

Therefore, by substitution, .

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: