DSQ: Calculating the length of the side of a right triangle - GMAT Quantitative

Knowing that the triangle has a right angle indicates the question can be solved using the Pythagorean Theorem, however it is unclear which side is the hypotenuse. For example, if 8 is not the hypotenuse, the length of the third side is 10. If 8 is the hypotenuse, the length of the third side is 5.3.

Additionally, if you only know that 8 is the longest side, the length of the third side could be anything greater than 2 and less than 8. Therefore, having both pieces of data will allow you to solve the problem.

The side opposite the angle of greatest measure is the longest of the three, so if we can determine which angle is the longest, we can answer this question.

It follows from Statement 1 by definition that , so, since the measures of the three angles total , , making right and the other two acute. This proves has the greatest measure of the three.

It follows from Statement 2 that is either right or obtuse; therefore, . Subsequently, the other two angles are acute, so again, has the greatest measure of the three.

From either statement alone, we can therefore identify as the side of greatest measure.

Statement 1: All we're given is the equation for finding the width, , which we'll use in the next statement.

Statement 2: Using the information from statement 1, we can set up an equation and solve for the length.

Statement 2 alone would not have provided sufficient information because we would have ended up with

and would not have been able to determine what the the values were.

Either statement alone is sufficient.

From either statement alone, it can be determined that and ; each statement gives one angle measure, and the other can be calculated by subtracting the first from , since the acute angles of a right triangle are complementary.

Also, since is the right angle, is the hypotenuse, and , opposite the angle, the shorter leg of a 30-60-90 triangle. From either statement alone, the 30-60-90 Theorem can be used to find the length of longer leg . From Statement 1 alone, has length times that of the hypotenuse, or . From Statement 2 alone, has length of the shorter leg, or .

From Statement 1 alone, the circumscribed circle has as its diameter the circumference divided by , or 20. From Statement 2 alone, the circle has area , so its radius can be found using the area formula;

The diameter is the radius doubled, which here is 20.

The hypotenuse of a right triangle is a diameter of the circle that circumscribes it, so the diameter of the circle gives us the length of the hypotenuse. However, we are looking for the length of a leg, . Either statement alone gives us only the length of the hypotenuse, which, without other information, does not give us any further information about the right triangle.

Since is given as the right angle of the triangle , we are being asked to evaluate the length of hypotenuse .

Statement 1 alone gives insufficient information. We note that the area of a right triangle is half the product of the lengths of its legs, and we examine two scenarios:

Case 1:

The area is

By the Pythagorean Theorem, hypotenuse has length

Case 2:

The area is

By the Pythagorean Theorem, hypotenuse has length

Both triangles have area 24 but the hypotenuses have different lengths.

Assume Statement 2 alone. A circle that circumscribes a right triangle has the hypotenuse of the triangle as one of its diameters, so the length of the hypotenuse is the diameter - or, twice the radius - of the circle. Since the area of the circumsctibed circle is , its radius can be determined using the area formula:

The diameter - and the length of hypotenuse - is twice this, or 10.

Statement 1 alone gives insufficient information. and , but it is not clear which of the three sides is the hypotenuse of . is not the longest side, so we know that or is the hypotenuse, and the other is the second leg. We explore the two possibilities:

If is the hypotenuse, then the legs are and ; since the lengths of the legs are 12 and 24, by the Pythagorean Theorem, has length

.

If is a leg, then the hypotenuse, being the longest side, is , and is the other leg; by the Pythagorean Theorem, has length

.

Statement 2 alone gives insufficient information in that it only gives information about the angles, not the sides.

Assume both statements are true. If is the hypotenuse and is a leg, then, since the hypotenuse measures twice the length of a leg from Statement 1, the triangle is 30-60-90, contradicting Statement 2. Therefore, by elimination, is the hypotenuse, and, consequently, .

Assume Statement 1 alone. Since we do not know whether is the hypotenuse or a leg of , we can show that can take one of two different values.

Case 1: If is the hypotenuse, then the legs are and ; since their lengths are 10 and 20, by the Pythagorean Theorem, has length

.

Case 2: If is a leg, then the hypotenuse, being the longest side, is , and is the other leg; by the Pythagorean Theorem, has length

.

Statement 2 gives insufficient information, since it only clues us in to the measures of the angles, not the sides.

Now assume both statements. Since one of the angles of the right triangle has measure , the other has measure ; the triangle is a 30-60-90 triangle, and therefore, its hypotenuse has twice the length of its shorter leg. Since, from Statement 1, , is the hypotenuse, and is the longer leg, the length of which, by the 30-60-90 Theorem, is times that of shorter leg , or .

Neither statement alone gives enough information to find , as each alone gives only one sidelength.

Assume both statements are true. While neither side is indicated to be a leg or the hypotenuse, the hypotenuse of a right triangle is longer than either leg; therefore, since and are of equal length, they are the legs. is the hypotenuse of an isosceles right triangle with legs of length 10, and by the 45-45-90 Theorem, the length of is times that of a leg, or .

The hypotenuse of a right triangle is its longest side. From Statement 1 alone, we can eliminate only as the hypotenuse, and from Statement 2 alone, we can eliminate only . From Statements 1 and 2 together, we can eliminate both, leaving as the hypotenuse.

Statement 1 alone gives insufficient information, as it only establishes congruence between two pairs of corresponding sides; without a third congruence or noncongruence between sides or angles, it cannot be established whether the triangles themselves are congruent. Statement 2 alone is insufficient, since it only compares two angles without giving any information about sidelengths, whether absolute or relative.

Assume both statements to be true. From Statement 1, we have the congruence statements and , and from Statement 2, we have that the included angle from has measure , and that the included angle from , being obtuse, has, by definition, measure greater than this. This sets up the conditions of the Side-Angle-Side Inequality Theorem, or Hinge Theorem, which states that in this situation, the third side opposite the greater angle is longer than the third side opposite the lesser. Therefore, it can be deduced that .

We are being asked to compare the lengths of the hypotenuses of the two triangles, since and are the sides opposite the right angles of their respective triangles.

Assume Statement 1 alone. We have that , , and , both being right angles, thereby establishing congruence between two pairs of sides and a pair of included angles. By the Side-Angle-Side Theorem, , and, consequently and have equal length.

Assume Statement 2 alone. We are only given information about the angle measures, but nothing about the lengths of the sides - actual lengths or comparisons. We can make no conclusions about which hypotenuse is longer.

The two statements together only give information about the angle measures of the two triangles. Without any information about the relative or absolute lengths of the sides, no comparison can be drawn between their hypotenuses.

We are being asked to compare the lengths of the hypotenuses of the two triangles, since and are the sides opposite the right angles of their respective triangles.

Statement 1 alone gives insufficient information, as shown by examining these two cases.

Case 1:

By the Pythagorean Theorem, the hypotenuse has length

The hypotenuse has length

Case 2:

The hypotenuse has length

and, as in Case 1, has length .

In both cases, and , so . But in the first case, was longer than , and in the second case, the reverse was true.

Statement 2 is insufficient in that it only gives us the congruence of one set of corresponding legs; without further information, it is impossible to determine which hypotenuse is longer.

Now assume both statements are true. Since and , by the subtraction property of inequality,

and

It follows from and that and ; by the addition property of inequality,

By the Pythagorean Theorem,

and

,

so the above inequality becomes, by substitution,

and

,

proving that is longer than .

Assume Statement 1 alone. Since and are the right angles of their respective triangles, and , the segments opposite the right angles, are their hypotenuses, and, subsequently, their longest sides. Specifically, . Since, from Statement 1, , it follows that .

Assume Statement 2 alone. Again, is the longest side of its triangle, so . But we cannot determine whether or without further information.

As each statement alone only gives the length of one segment, neither statement alone is sufficient to find the length of any other segment, including, in particular, .

Assume both statements to be true. can be found by way of the Pythagorean Theorem:

Now note that is the altitude of the right triangle from the vertex of the right angle, which divides the right triangle into two triangles similar to each other and the large triangle. Specifically,

By similarity,

.

Assume Statement 1 alone. The hypotenuse of a right triangle is longer than either of its two other sides; since and are equal in length, neither is the hypotenuse. This leaves as the hypotenuse.

Assume Statement 2 alone. Since , neither can be the right angle. Therefore, is the right angle, and its opposite side, , is the hypotenuse.

Since we are comparing angles, we need to identify the angle of greatest measure; in a right triangle, the angle of greatest measure is the right angle, and the side opposite it is the hypotenuse.

Statement 1 is insufficient, since we can eliminate only angle as the right angle, and, subsequently, only as the hypotenuse. Similarly, Statement 2 is insufficent, since we can eliminate only angle as the right angle, and, subsequently, only as the hypotenuse. But if we are given both statements, we can eliminate and as the hypotenuse, leaving as the hypotenuse.

In a right triangle, the angle of greatest measure is the right angle, and the side opposite it is the hypotenuse.

Assume both statements are true. We can eliminate as the right angle, as it has measure less than both and . However, we have no information that tells us which of and has the greater measure, so we cannot determine which is the right angle. Subsequently, we cannot eliminate either of their opposite sides, or , respectively, as the hypotenuse.

The side opposite of the right angle is the hypotenuse. Statement 1 alone eliminates as the right angle, and, as a consequence, as the hypotenuse - but only .

From Statement 2 alone, we have that , meaning that

and

Since is shorter than , , and only , is eliminated as the hypotenuse.

If both statements are assumed to be true, however, then both and can be eliminated as the hypotenuse, leaving as the only choice.