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Example Questions
Example Question #1 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle
Given equilateral triangle
and right triangle , which, if either, is longer, or ?Statement 1:
Statement 2:
is a right angle.EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Assume Statement 1 alone. Since all three sides of
are congruent - specifically, - and , it follows by transitivity that . However, no information is given as to whether has length greater then, equal to, or less than , so which of and , if either, is the longer cannot be answered.Assume Statement 2 alone. Since
is the right angle of , is the hypotenuse and this the longest side, so and . However, no comparisons with the sides of can be made.Now assume both statements are true.
as a consequence of Statement 1, and as a consequence of Statement 2, so .Example Question #4 : Equilateral Triangles
What is the length of side
of equilateral triangle ?Statement 1:
is a diagonal of Rectangle with area 30.Statement 2:
is a diagonal of Square with area 36.BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
An equilateral triangle has three sides of equal measure, so if the length of any one of the three sides can be determined, the lengths of all three can be as well.
Assume Statement 1 alone.
is a diagonal of a rectangle of area 30. However, neither the length nor the width can be determined, so the length of this segment cannot be determined with certainty.Assume Statement 2 alone. A square with area 36 has sidelength the square root of this, or 6; its diagonal, which is
, has length times this, or . This is also the length of .
Example Question #5 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle
What is the length of side
of equilateral triangle ?Statement 1:
, , and are all located on a circle with area .Statement 2: The midpoints of all three sides are located on a circle with circumference
.BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
We demonstrate that either statement alone yields sufficient information by noting that the circle that includes all three vertices of a triangle - described in Statement 1 - is its circumscribed circle, and that the circle that includes all three midpoints of the sides of an equilateral triangle - described in Statement 2 - is its inscribed circle. We examine this figure below, which shows the triangle, both circles, and the three altitudes:
The three altitudes intersect at
, which divides each altitude into two segments whose lengths have ratio 2:1. is the center of both the circumscribed circle, whose radius is , and the inscribed circle, whose radius is .Therefore, from Statement 1 alone and the area formula for a circle, we can find
from the area of the circumscribed circle:
From Statement 2 alone and the circumference formula for a cicle, we can find
from the circumference of the inscribed circle:
By symmetry,
is a 30-60-90 triangle, and either way, , and .Example Question #5 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle
Given two equilateral triangles
and , which, if either, is greater, or ?Statement 1:
Statement 2:
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
An equilateral triangle has three sides of equal length, so
and .Assume Statement 1 alone. Since
, then, by substitution, .Assume Statement 2 alone. Since
, it follows that , and again by substitution, .Example Question #7 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle
You are given two equilateral triangles
and .Which, if either, is greater,
or ?Statement 1: The perimeters of
and are equal.Statement 2: The areas of
and are equal.Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Assume Statement 1 alone, and let
be the common perimeter of the triangles. Since an equilateral triangle has three sides of equal length, and , so .Assume Statement 2 alone, and let
be the common area of the triangles. Using the area formula for an equilateral triangle, we can note that:
and ,
so
.
Example Question #11 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle
Given equilateral triangle
and right triangle , which, if either, is longer, or ?Statement 1:
Statement 2:
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Assume Statement 1 alone. Since all three sides of
are congruent - specifically, - and , it follows by transitivity that . However, no information is given as to whether has length greater than, equal to, or less than , so it cannot be determined which of and , if either, is the longer. By a similar argument, Statement 2 yields insufficient information.Now assume both statements are true.
and are each congruent to one of the congruent sides of equilateral and are therefore congruent to each other. However, the hypotenuse of a right triangle must be longer than both legs, so the hypotenuse of is . is also longer than any segment congruent to one of the legs, which includes all three sides of - specificially, is longer than .Example Question #12 : Equilateral Triangles
is equilateral. may or may not be equilateral.
which, if either, is longer,
or ?Statement 1:
Statement 2:
andBOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Assume Statement 1 alone.
is equilateral, so . Also, by the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third, so . From Statement 1, , so by substitution, , and .Statement 2 alone provides insufficient information. For example, assume
is an equilateral triangle with sidelength 9. If is an equilateral triangle with sidelength 8, the conditions of the statement hold, and . However, if is a right triangle in which , , and , the conditions of the statement still hold, but .Example Question #13 : Equilateral Triangles
Given equilateral triangles
and , which, if either, is longer, or ?Statement 1:
Statement 2:
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
All sides of an equilateral triangle have the same measure, so we can let
be the common sidelength of , and be that of .Statement 1 can be rewritten as
; Statement 2 can be rewritten as . The equivalent question is whether we can determine which, if either, is greater, or . The two statements together are insufficient to answer the question, however; 5 and 10 have sum 15 and product 50, but we cannot determine without further information whether and , or vice versa. Therefore, we do not know for sure whether a side of is longer than a side of - specifically, which of or is longer.Example Question #12 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle
Given equilateral triangles
and , which, if either, is longer, or ?Statement 1:
Statement 2:
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
All sides of an equilateral triangle have the same measure, so we can let
be the common sidelength of , and be that of .Statement 1 can be rewritten as
; Statement 2 can be rewritten as . The equivalent question is whether we can determine which, if either, is greater, or .Statement 1 alone yields insufficient information; for example, the two numbers added together could be 10 and 14, but it is impossible to determine whether
or is the greater of the two. Statement 2 alone is also insufficient, for a similar reason; for example, the two numbers could be 9 and 16, but again, either or could be the greater.Now assume both statements. The only two numbers that can be added to yield a sum of 24 and multiplied to yield a product of 144 are 12 and 12; therefore,
, and and have the same sidelengths. Specifically, and have the same length.Example Question #1 : Dsq: Calculating The Area Of An Equilateral Triangle
What is the area of
?(1) The height
is 5.(2) The base
is 4.Statement 2 alone is sufficient
Each statement alone is sufficient
Statements 1 and 2 together are not sufficient
Both statements taken together are sufficient
Statement 1 alone is sufficient
Both statements taken together are sufficient
To find an area of a triangle we need the length of the height and the length of the corresponding basis.
Each statement 1 and 2 alone is not sufficient, since we don't know whether the triangle is equilateral. Indeed, we need to take both statements to be able to calculate the area.
Hence, both statements together are sufficient.
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