All SAT II Math II Resources
Example Questions
Example Question #8 :Finding Sides
Regular Pentagon周长35。hasas its midpoint; segmentis 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, sinceis the midpoint of,.
Also, each interior angle of a regular pentagon measures.
Below is the pentagon in question, withindicated andconstructed; all relevant measures are marked.
A triangleis formed with,, and included angle measure. The length of the remaining side can be calculated using the law of cosines:
whereandare the lengths of two sides,is the measure of their included angle, andis the length of the third side.
Setting, and, substitute and evaluate:
;
Taking the square root of both sides:
,
the correct choice.
Example Question #9 :Finding Sides
Regular Hexagonhas perimeter 360.andhaveandas midpoints, respectively; segmentis drawn. To the nearest tenth, give the length of.
The perimeter of the regular hexagon is 360, so each side measures one sixth of this, or 60. Sinceis the midpoint of,.
同样的,.
Also, each interior angle of a regular hexagon measures.
Below is the hexagon with the midpointsand, and withconstructed. Note that perpendiculars have been drawn tofromand, with feet at pointsandrespectively.
是一个矩形,所以呢.
.
This makesandthe short leg and hypotenuse of a 30-60-90 triangle; as a consequence,
.
For the same reason,
Adding the segment lengths:
.
Example Question #10 :Finding Sides
Regular Pentagonhas 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 triangleis formed with, and included angle measure. The length of the remaining side can be calculated using the Law of Cosines:
whereandare the lengths of two sides,the measure of their included angle, andthe length of the side opposite that angle.
Setting, and, substitute and evaluate:
Taking the square root of both sides:
,
the correct choice.
Example Question #51 :Geometry
Given a cube, if the volume is 100 feet cubed, what must be the side?
Write the formula for the volume of the cube.
To solve for, cube root both sides.
Substitute the volume.
The answer is:
Example Question #1 :Finding Angles
The angles containing the variableall reside along one line, therefore, their sum must be.
Becauseandare opposite angles, they must be equal.
Example Question #2 :Finding Angles
What angle do the minute and hour hands of a clock form at 6:15?
There are twelve numbers on a clock; from one to the next, a hand rotates. At 6:15, the minute hand is exactly on the "3" - that is, on theposition. The hour hand is one-fourth of the way from the "6" to the "7" - that is, on theposition. Therefore, the difference is the angle they make:
.
Example Question #3 :Finding Angles
In triangle,and. Which of the following describes the triangle?
is acute and isosceles.
is obtuse and scalene.
is acute and scalene.
is obtuse and isosceles.
None of the other responses is correct.
is acute and isosceles.
Since the measures of the three interior angles of a triangle must total,
All three angles have measure less than, making the triangle acute. Also, by the Isosceles Triangle Theorem, since,; the triangle has two congruent sides and is isosceles.
Example Question #4 :Finding Angles
In,andare complementary, and. Which of the following is true of?
is acute and isosceles.
is acute and scalene.
is right and isosceles.
is right and scalene.
None of the other responses is correct.
is right and scalene.
andare complementary, so, by definition,.
Since the measures of the three interior angles of a triangle must total,
is a right angle, sois a right triangle.
andmust be acute, so neither is congruent to; also,andare not congruent to each other. Therefore, all three angles have different measure. Consequently, all three sides have different measure, andis scalene.
Example Question #5 :Finding Angles
The above figure is a regular decagon. Evaluate.
As an interior angle of a regular decagon,measures
.
Sinceandare two sides of a regular polygon, they are congruent. Therefore, by the Isosceles Triangle Theorem,
The sum of the measures of a triangle is, so
Example Question #6 :Finding Angles
The above hexagon is regular. What is?
None of the other responses is correct.
Two of the angles of the quadrilateral formed are angles of a regular hexagon, so each measures
.
The four angles of the quadrilateral are. Their sum is, so we can set up, and solve for在方程:
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