SAT Math › Finding Sides
Which of the following describes a triangle with sides one kilometer, 100 meters, and 100 meters?
The triangle cannot exist.
The triangle is acute and equilateral.
The triangle is obtuse and isosceles, but not equilateral.
The triangle is acute and isosceles, but not equilateral.
The triangle is obtuse and scalene.
One kilometer is equal to 1,000 meters, so the triangle has sides of length 100, 100, and 1,000. However,
That is, the sum of the least two sidelengths is not greater than the third. This violates the Triangle Inequality, and this triangle cannot exist.
The above figure is a regular pentagon. Evaluate to the nearest tenth.
Two sides of the triangle formed measure 4 each; the included angle is one angle of the regular pentagon, which measures
The length of the third side can be found by applying the Law of Cosines:
where :
Note: figure NOT drawn to scale.
Refer to the above diagram.
.
Which of the following expressions is equal to ?
By the Law of Sines,
.
Substitute ,
, and
:
We can solve for :
Note: figure NOT drawn to scale.
Refer to the triangle in the above diagram.
Evaluate . Round to the nearest tenth, if applicable.
By the Law of Cosines,
Substitute :
Note: figure NOT drawn to scale.
Refer to the triangle in the above diagram.
Evaluate . Round to the nearest tenth, if applicable.
By the Law of Cosines,
Substitute :
Regular Pentagon has perimeter 60.
To the nearest tenth, give the length of diagonal .
The perimeter of the regular pentagon is 60, so each side measures one fifth of this, or 12. Also, each interior angle of a regular pentagon measures .
The pentagon, along with diagonal , is shown below:
A triangle is formed with
, and included angle measure
. The length of the remaining side can be calculated using the Law of Cosines:
where and
are the lengths of two sides,
the measure of their included angle, and
the length of the side opposite that angle.
Setting , and
, substitute and evaluate
:
Taking the square root of both sides:
,
the correct choice.
The above figure is a regular decagon. Evaluate to the nearest tenth.
Two sides of the triangle formed measure 6 each; the included angle is one angle of the regular decagon, which measures
.
Since we know two sides and the included angle of the triangle in the diagram, we can apply the Law of Cosines,
with and
:
is a rhombus with side length
. Diagonal
has a length of
. Find the length of diagonal
.
A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.
Thus, we can consider the right triangle to find the length of diagonal
. From the problem, we are given that the sides are
and
. Because the diagonals bisect each other, we know:
Using the Pythagorean Theorem,
is rhombus with side lengths in meters.
and
. What is the length, in meters, of
?
5
12
15
24
30
A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.
Thus, we can consider the right triangle to find the length of diagonal
. From the given information, each of the sides of the rhombus measures
meters and
.
Because the diagonals bisect each other, we know:
Using the Pythagorean theorem,
Regular Pentagon has perimeter 35.
has
as its midpoint; segment
is drawn. To the nearest tenth, give the length of
.
The perimeter of the regular pentagon is 35, so each side measures one fifth of this, or 7. Also, since is the midpoint of
,
.
Also, each interior angle of a regular pentagon measures .
Below is the pentagon in question, with indicated and
constructed; all relevant measures are marked.
A triangle is formed with
,
, and included angle measure
. The length of the remaining side can be calculated using the law of cosines:
where and
are the lengths of two sides,
is the measure of their included angle, and
is the length of the third side.
Setting , and
, substitute and evaluate
:
;
Taking the square root of both sides:
,
the correct choice.