Acute / Obtuse Triangles - GMAT Quantitative

By the Triangle Inequality, the sum of the lengths of two shortest sides must exceed that of the third.

Case 1: is the greatest of the three sidelengths.

Then

Case 2: is not the greatest of the three sidelengths - that is, 11 is.

Then , or, equivalently, .

Therefore, .

The triangle can exist by the Triangle Inequality, since the sum of the two smaller sides exceeds the greatest:

To determine whether the triangle is acute, right, or obtuse, add the squares of the two smaller sides, and compare the sum to the square of the largest side.

Since this sum is greater, the triangle is acute.

If these are the measures of the interior angles of a triangle, then they total . Add the expressions, and solve for .

One angle measures The others measure:

Since the largest angle measures greater than , the angle is obtuse, and the triangle is as well. Since the three angles each have different measure, their opposite sides do also, making the triangle scalene.

The three sides of a scalene triangle have different measures, so 15 can be eliminated.

By the Triangle Inequality, the sum of the lengths of the two smaller sides must exceed the length of the third side. Since , 8 violates this theorem; since , 22 does as well.

10 is a valid measure of the third side, since ; it makes all three segments of different length, so it is the correct choice.

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.

By trial and error, we get four ways to add distinct primes to yield sum 33:

In each case, however the Triangle Inequality is violated - the sum of the two shortest lengths does not exceed the third.

No triangle can exist as described.

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.

A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is a prime integer.

One of the sides cannot be 2, since the sum of 2 and two odd primes would be an even number greater than 2, a composite number. Therefore, beginning with the least three odd primes, add increasing triples of distinct prime numbers, as follows, until a solution presents itself:

- incorrect

- correct

The correct answer, 19, presents itself quickly.

A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is 47.

There are ten ways to add three distinct primes to yield sum 47:

By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate all but four:

The greatest possible length of the longest side is 23.

The height of the obtuse isosceles triangle bisects the angle and forms two congruent right triangles. The hypotenuse of each of these triangles is either side of equivalent length, and we can see that the base of either triangle makes up half of the hypotenuse of the obtuse isosceles triangle. Because we know the angle opposite each base is half of , or , we can use the sine of this angle to find the length of the base. As there are two congruent right triangles that make up the obtuse isosceles triangle, the length of either base makes up half of the overall hypotenuse, so we then multiply the result by to obtain the final answer. In the following solution, is the length of the base of one of the right triangles, is the length of the two equivalent sides, and is the length of the hypotenuse:

Use Heron's formula:

where , and

In this problem, the base is 12 and the height is 6. Therefore:

The figure is a composite of a rectangle and a triangle, as shown:

The rectangle has area

The triangle has area

The total area of the figure is

This can be illustrated by showing this triangle inscribed inside a rectangle whose vertices are :

The area of the white triangle is the one whose area we calculate. To do this, we need the area of the square:

The area of the red triangle:

The area of the green triangle:

And the area of the beige triangle:

The area of the white triangle will be as follows:

The only restriction on the measure of the vertex angle of an isosceles triangle is the restriction on any angle of a triangle - that it fall between and , noninclusive. If is any number in that range, each base angle, the two being congruent, will measure , which will fall in the acceptable range.

Since all of these measures fall in that range, the correct response is that all are allowed.

The easiest way to solve this is to graph the three lines and to observe the dimensions of the resulting triangle. It helps to know the coordinates of the three points of intersection, which we can do as follows:

The intersection of and the -axis - that is, the line can be found with some substitution:

The lines intersect at

The intersection of and the -axis can be found the same way:

These lines intersect at

The intersection of and can be found via the substitution method:

The lines intersect at

The triangle therefore has these three vertices. It is shown below.

As can be seen, it is a triangle with base 9 and height 12, so its area is

The vertical segment connecting and can be seen as the base of this triangle; this base has length . The height is the perpendicular (horizontal) distance from to this segment, which is 6, the same as the -coordinate of this point. The area of the triangle is therefore

.

We can view the horizontal segment connecting , and as the base; its length wiill be . The height will be the perpendicular (vertical) distance to this segment from the opposite point , which is , the -coordinate; therefore, the area of the triangle will be half the product of these two numbers, or

.

If two sides are 4 and 8, then the third side must be greater than and less than . This means can be 5, 6, 7, 8, 9, 10, or 11.