Calculating the height of an acute / obtuse triangle - GMAT Quantitative

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 , 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.

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 ; 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 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 .

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:

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 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 .

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,

.

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 , 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.

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 ; 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 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 .