Review
e-mail joan_wing@pinecrest.edu
Be able to find missing lengths in 30-6-90 and 45-45-90 triangles.
Be able to find and name the parts of a regular right prism or pyramid.
d. Vertices e. Base edge f. Height of the prism
Ans. a Triangles AEB and DFC, b) any of the rectangles DFAE, DEBC, BAFC c. DE, BC, FA d) Pts A,B,C,D,E,F e) Any of the sides of the triangular bases , AE, BE, BA, DF, FC, FD f) DE, AF or BC
d. Base edge e. slant height f. Height of the pyramid
Ans. a) FEBC b. Any of the triangular sides DEB, DBC, DFC, or DEF c) DB, DC, DF, DE e) DA f) DG
Be able to use the trigonometric functions of sine, cosine and tangent to find missing measures in a right triangle.
Ans. Use the trig function of Tan to solve for x. Tan 65 = 8/x When x is isolated it = 8/tan65 = 8/2.1445 = 3.7 To find y, use the sine function Sin 65 = 8/y. y= 8/sin 65 = 8/.9063 = 8.8
Ans. To find the angle the inverse trig function must be used. Since the tan A = 5/15 use tan-1(5/15) = 18o
Use the properties of chords and their relationships to radii and tangents to find missing lengths of segments in a circle.
Use the properties of chords and their relationships to radii and tangents to find missing lengths of segments in a circle.
8. If O is the center of the circle and OA is perpendicular to XY, and XY=48, when OA = 7, find the radius of the circle.
Ans. Draw in OX as the radius, XA = 24 (half of 48) and OA=7 Use the Pythagorean theorem to solve for OX c2= a2 + b2 , c2= 242 + 72, c2= 576 + 49 = 625 c = 25
9. Find the length of a chord that is 8 inches from the center of a circle with a diameter of 34.
Ans. 30, Radius of the circle = 1/2 diameter = 17, 8 is the perpendicular to the chord which splits the chord in half. Use the pythag to find x, then double to find the chord. 172= 82 + x2 , 289 =64 + x2 , 225 =x2 , x=15, 2x=30
Ans.
Let x = AO and OC lengths since radii of a circle are congruent, Radii
drawn to tangents at their point of tangents are perpendicular. Use the
pythag on the right triangle formed. (x+4)2= x2 +
82, x2+8x+16= x2 + 64, 8x+16= 64,
8x=48, x=6
If AB is a common tangent to circles O and P and AO = 10 , BP = 17 and the circle centers are 25 cm apart, find the length of the common tangent
Ans. 24 - Create a rectangle by dropping a perpendicular from the center of the smaller circle to the radius of the larger. Since opposite sides of a rectangle are congruent, one side is 10 and the other x, This leaves 7 units for the side of the right triangle formed. We already know the hypotenuse is 25 since the circle centers are 25 units apart. Using pythag , 252= x2 + 72, 625=x2 +49 , 576=x2 , x = 24
12. The ratio of the measures of arc GI to JH is 3:2, and the measure of the angle formed by the two chords is 65, find the measure of arc GI
Ans. 52 - Since an angle formed by two chords intersecting in a circle = 1/2 the sum of the intercepted arcs, let GI = 2x and JH = 3x. Set up the equation (2x+3x)/2=65, 5x/2=65 5x=130, x=26. If x=26 the GI = 2x or 2(26)=52
13.
Ans. 44- Since an angle formed by two tangents = 1/2(subtraction of the intercepted arcs) let NF = x and set up the equation (98-x)/2 = 27, 98-x= 54, x = 44
14.
Ans. 108- The two arcs of the circle add up to 360, 4x + x = 360 5x = 360, x = 72, Since when two tangents are drawn to a circle, the inner arc and the angle formed are supplements, solve for < P by subtracting 72 from 180.
15a. Find the measure of x.
Ans. 25 - When a quadrilateral is inscribed in a circle, the opposite angles are supplementary. Set up the equation 4x-15 +2x+45 = 180, 6x+30 = 180, 6x=150, x=25
Use the power theorems and two tangent theorem to find lengths of segments in relationship to a circle.
15b)
Ans. 4 - Let x = BD, since tangents drawn to a circle from the same point are congruent, BE=x, Since AB = 12, represent AE as 12-x and AF will also be 12-x. BC = 10 so represent CD and CF as 10-x, Since AC = 14, AF + FC = 14 or 10-x +12-x = 14. 22-2x = 14, -2x=-8, x = 4
16.
Ans. 8 or 12, Since when two chords intersect in a circle the product of the segments on one chord = the product of the segments on the other, let PX = x and XT = 20 - x, Set up the equation (16)(6) = x ( 20 - x ) 96 = 20x - x2 , x2 -20x +96 = 0 , (x-12)(x-8) = 0 x = 12 or x=8.
Understand the meaning of inscribed and circumscribed and find measures given certain information concerning the figures.
17. If a circle is circumscribed about a square with an area of 49 sq. cm, find the circumference of the circle.
ans.
18. If an equilateral triangle is inscribed in a circle and the perimeter of the triangle is 18, find the radius of the circle.
ans.
Find the areas of plane figures and shaded regions.
19. Find the area of an equilateral triangle with a perimeter of 30
ans.
20. Find the area of an isosceles triangle with legs of 10 cm and a base of 12 cm.
ans. 48
21. Find the area of an isosceles trapezoid with bases of 8 cm and 26 cm and legs = 15 cm.
ans. 204 Drop two perpendiculars to the base of the trapezoid. A rectangle is formed with 18 left over of the 26 units to be split for the two end pieces . Use the Pythagorean Theorem on the right triangles formed to find the height of the trapezoid 152= x2 + 92 , height =12. Use the formula for the area of a trapezoid A = 1/2 h (b1 + b2) = 1/2 (12)(26+8) = 6 ( 34) =204
22. Find
the area of a hexagon with an apothem of
ans.
When the apothem and radius are drawn in a regular hexagon, a 30-60-90 triangle
is formed to get back from the side opposite the 60 degree angle, divide by root
3. This gives 8 for the short leg and the side of the hexagon will measure
16. To find the area of any regular polygon, the Area =
1/2(apothem)(perimeter)
23. Find the area of a rhombus with a perimeter of 52 and a diagonal of 10.
ans. 120 If the perimeter =52, each side must be 13 since a rhombus has 4 equal sides. The diagonals of a rhombus are perpendicular bisectors of each other so the diagonal of 10 is split into 5 and 5. Use the pythagorean theorem on the right triangle and solve for x which = 12. The area of a rhombus is found by A = 1/2 diagonal x diagonal or 1/2 (10)(24) = 120
24.
(I can't figure out how to shade the areas - check your sheet - it is areas bounded by FA,AE and arc FE and EC,CD and arc ED-or area outside of the square)
ans.
Find the area of the square and subtract it from the area of the sector.
Since the side of the square is 8, the diagonal of the square must be
which is the radius of the sector. To find the area of the sector use the
formula
then subtract 64 from this and
you have the shaded region.
25.
ans.
To find the area of the shaded region find the area of the sector and subtract
the area of the triangle. Area of the sector is found by:
To find the
area of the triangle = drop a perpendicular from J to KI which will split the
120 degree angle into two 60 and form two 30-60-90 triangles. The height of the
triangle will be 5 and the base 
Since the area of a triangle = 1/2 b h then the triangles area =
26. If the ratio of the sides of two similar trapezoids is 5 to 6 and the area of the smaller trapezoid is 300 sq units, find the area of the larger.
ans. 432 Since areas of similar figures compare as squares of the ratios of the sides the area ratio is 25/36. Set up the proportion 25/36 = 300/x and solve for x by cross multiplication. 25x = 10800 x = 432
Find the lateral area, total area and volume of prisms, cylinders, cones, spheres, and pyramids.
27. Given a right square pyramid with base edge of 12 and slant height of 10, find the LA, TA and Volume.
ans. LA = 240, TA = 384, V= 384
To find the LA use the formula LA = 1/2 pl , LA = 1/2(48)(10)= 240
To find the total area add on the area of the square base which is a 12 x 12 square = 144
TA = 240 + 144 = 384
Volume is found by using the
formula V = 1/3 Bh or 1/3 the area of the base times the height of the
pyramid. The height must be found by using the Pythagorean theorem on the
right triangle formed by half the base edge, the slant height and the height of
the pyramid. 
28. If
the volume of a sphere is 288
cubic units, find the surface area of the sphere.
ans.
Use the formula for volume and solve for the radius, then substitute into the
formula for a spheres surface area the value of the radius.
29. Find the surface area and volume if the central area is a right cylinder and the ends are a hemisphere and a right cone.
ans.
Surface area = 
, Volume =
( I think I had this one wrong on my answer key in class so check it carefully
to make sure I haven't made an error)
Find the area of the hemisphere using half the surface area of the sphere, find the LA of the cylinder, and find the LA of the cone to find the surface area of the entire figure.
To find the volume of the solid, find the volume of the hemisphere, volume of the cylinder and volume of the cone and total them. (Note you need to use the pythagorean theorem on the cone to find its height when calculating the volume)
29. Find the LA , TA and Volume of the cone if the radius is 5 and the height is 12.
ans.
Use the pythagorean theorem to find the slant height which comes out 13.
Answers to Multiple Choice Practice
1. B 2. D (make sure you change the angle across from h to 30 degrees) 3. D 4. C 5. B 6. A 7. C 8. C 9. A 10. D
11. B 12. D 13. D 14. C 15. A 16. C 17. C 18. C 19. D 20. A 21. D 22. B 23. A 24. A 25. C
26. B 27, A 28. A 29. A
EXAM -
Findeiss
Thursday, June 3th
1:00 BRING CALCULATORS, AND NUMBER TWO PENCILS
It has been a pleasure and a privilege teaching all of you this year. Have a great summer and I'll look forward to seeing you back at PC in the fall!!!!