Math 5  Trigonometry  Chapter 1 Test                  Name__________________________

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

 

1.      What angle has the same measure as its complement?  How do you know?

2.      What is the complement of the supplement of 100˚ angle? 

3.      Given Quadrilateral ABCD with diagonal BD forming congruent angles  and , what kind of quadrilateral do you think this is?  Why?  Give as persuasive a justification as you can.

 

4.      Find the perimeter of an isosceles triangle with base = 12cm and height = 4cm.

5.      Find the area of an isosceles triangle with sides of length 2cm, 3cm and 3cm.

6.      Given that AC = AB and AE = ED = DB = BC in  at right, find the degree measure of

7.      Carlos and Karla start at the north west corner of a square block measuring 120 meters on a side and Carlos starts walking around the block by heading east at 0.5 meters per second at the same time that Karla starts walking around the block by heading south at 0.6 meters per second.  What is the length of the line segment connecting Carlos and Karla after five minutes?

8.      Consider a 120˚ sector of a circle with radius 10cm.

a.       Find the perimeter of the sector.

b.      Find the area of the sector.

9.      In the diagram at right, AB is a diameter of the circle centered at O and C is a point on the perimeter of the circle.

a.       Express  in terms of  .

b.      Explain why  is isosceles.

c.       Express  in terms of  .

d.      Express  in terms of  .

10.  In the figure at right,

a.       Show that  

b.      Find the length of AE.

c.       Draw BD.  Is ?  Why or why not?

 

 

 

Solutions

 

1. If x is its own complement the x + x = 90˚, or x = 45˚.
2. The supplement of 100˚ is 80˚ whose complement is 10˚.
3. We know that if a transversal cuts two parallel lines, alternate interior angles are congruent.  It seems reasonable that the converse is also true: if the alternate interior angles are congruent, then the lines are parallel.  Thus AB and CD are parallel and AD and BC are parallel, so quadrilateral ABCD is a parallelogram.
4. An isosceles triangle with base = 12cm and height = 4cm has equal sides of length =
   
   .  Thus the perimeter is
   
    cm.
5. The height of an isosceles triangle with sides of length 2cm, 3cm and 3cm is
   
   , so its area is
   
    cm².
6. Let  x =
   
    then, since
   
    is isosceles with base AD,
   
    and since the sum of interior angles is 180˚,
   
   .  Further, since
   
    is a straight angle,
   
    is the supplement of
   
   .  But
   
    is isosceles, so
   
    and thus (interior angles’ sum is 180˚)
   
   .  Now since
   
    is a straight angle,
   
    and, substituting,
   
    whence
   
    Also,
   
    is isosceles so
   
   .  Finally, since
   
    is isosceles,
   
    so we have the equation  
   

7.      Since 5 minutes is 300 seconds, Carlos will travel (0.5 m/s)*300s = 150m, putting him at a point 30 m south of the north edge on the east side of the block.  Karla will travel (0.6 m/s)*300s = 180m, so that she’s on the south edge, 60m east of the west side. Thus the distance between them is the hypotenuse of a right triangle with legs 60 and 90, D =  m

8. Consider a 120˚ sector of a circle with radius 10cm.

a.       The perimeter of the sector is  

b.      The area of the sector is  cm²

9. In the diagram at right, AB is a diameter of the circle centered at O and C is a point on the perimeter of the circle

a.       .

b.      AD and DC are radii, so AD = DC.

c.        

d.       

10. 
    
     so
    
     and thus
    
    . 
    So CE/12 = 4/5 and CE = 9.6 whence AE = 3 + 9.6 = 12.6.
    BD = 13 and 13/12 is not equal to 5/4 so
    
     is not similar to