Math 5  Trigonometry  Chapter 1 Test                  Name__________________________

Show all work for credit.  Write all responses on separate paper.



1.      Explain why the two acute angles of a right triangle are complementary.

2.      What is the degree measure of angle x in the figure at right?  Explain how you know.

3.      True or false: if one side of a quadrilateral is congruent to the opposite side, then the quadrilateral is either a parallelogram or an isosceles trapezoid.  Justify your answer.  

4.      In the isosceles right triangle shown at right,
AB = 20.

(a) What is the length of AC?

(b)  Draw the altitude from  
       C to AB.  What is its
       length?

 

5.      Two boats leave a dock at the same time and at a 90o angle from each other. After 3 hours one boat is 10 miles from the dock, while the other is 40 miles from the dock. How far are the boats from each other?  Write your answer in simplest radical form.

6.      Find the (a) perimeter and (b) area of a rectangle with one side 10 cm and diagonal 13 cm. 

7.      Explain why a diagonal of a parallelogram creates two congruent triangles.

8.      Find the area of region ACDB.   = 9cm and CD = 11cm are concentric arcs with center O .

9.      A closed right circular cylinder has a radius of 3 meters. Find the volume of the cylinder if its lateral surface area is 84π square meters. Leave your answer in terms of π.

 

 

10.  Show that triangles BCD and ACE are similar and use that similarity to find the length of DE in the diagram at right.


Math 5  Fall ’07  Chapter 1 Test Solutions

1.      Explain why the two acute angles of a right triangle are complementary.
ANS:  The sum of the interior angles of any triangle is 180°.  In a right triangle, one angle is 90° and that means the remaining angles add up to 90°, which means they’re complementary.

2.      What is the degree measure of angle x in the figure at right?  Explain how you know
ANS: Since ΔABC is isosceles,  = 58°, and since , x must be supplementary to the corresponding angle at C and thus x = 122°.

3.      True or false: if one side of a quadrilateral is congruent to the opposite side, then the quadrilateral is either a parallelogram or an isosceles trapezoid.  Justify your answer.  
ANS: False.  Consider the quadrilateral ABCD with vertices at A(0,0), B(2,0), C(5,4) and D(0,5), as shown at right. 
In the figure, AD = BC = 5, but the figure is neither a parallelogram nor an isosceles trapezoid.

4.      In the isosceles right triangle shown at right, AB = 20.
(a) What is the length of AC?
ANS:  AC/AB = AC/20 =  so

(b)  Draw the altitude from C to AB.  What is its length?
NS:  This line would create two congruent isosceles right triangles each half the size of the original.  Thus the altitude would be 10.

5.      Two boats leave a dock at the same time and at a 90o angle from each other. After 3 hours one boat is 10 miles from the dock, while the other is 40 miles from the dock. How far are the boats from each other?  Write your answer in simplest radical form.
ANS:  The boats’ paths are legs of a right triangle and the distance between them is the hypotenuse of that triangle:

6.      Find the (a) perimeter and (b) area of a rectangle with one side 10 cm and diagonal 13 cm. 
ANS: The short side of the rectangle is  so the perimeter is  and the area is

7.      Explain why a diagonal of a parallelogram creates two congruent triangles
ANS:  The transversal AC forms alternate interior angles congruent so that .  Also, AC is congruent to itself. This establishes the conditions of the ASA so we can conclude the triangles are congruent.

8.      Find the area of region ACDB.   = 9cm and CD = 11cm are concentric arcs with center O.
ANS: Solve the arc length formula  for the radius and plug in the values of s:  and  and then use these in the formula  for the area of a sector to compute the difference of the sectors’ areas:  cm2.

9.      A closed right circular cylinder has a radius of 3 meters. Find the volume of the cylinder if its lateral surface area is 84π square meters. Leave your answer in terms of π.
ANS:  The formula for lateral surface area is 2πrh.  Substituting r = 3 and setting this equal to 84π we have 6πh = 84π whence h = 14, thus the volume is . 

10.  Show that triangles BCD and ACE are similar and use that similarity to find the length of DE in the diagram.

ANS: Since a transversal cutting parallel lines makes corresponding angles congruent,  is a right angle. Also triangles BCD and ACE share an angle at A.  Since two of the angles are equal, and the sum of interior angles is 180°, it must be that the third angles are also equal and thus the triangles are equiangular, which also means they are similar.  Let x = ED.  Then, since corresponding parts of similar triangles are proportional, .