Recall: in the figure below, d is parallel to f. Then, the angles formed by intersecting them with a third line are congruent, namely A1 is congruent with B1, A2 with B2 and so on.
Measuring
Triangles, Areas and
Volumes
Recall: in the figure below, d is parallel to f.
Then, the angles
formed by intersecting them with a third line are congruent, namely A1 is
congruent with B1, A2 with B2 and so on.
Triangles.The Trigonometry of the Triangle
An
oblique triangle is any triangle that is not a right triangle.
To
solve
an oblique triangle you
need to know the measure of at
least one side and any two other parts of
the triangle- two sides, two
angles, or one angle and one side. This breaks
down into the following
four cases.
Area of
a
Triangle
Parallelogram
Trapezoid
Rectangle
Square
Rhombus
Circle
Sector of
Circle
Segment of Circle
Cube
Cuboid
Right Circular Cylinder
Cone
Sphere
Pyramid
Homework:
1. AB is a line 652 feet long on one bank of a stream, and C is a point on the opposite bank. A = 53° 18', and B = 48° 36'. Find the width of the stream from C to AB.
2. In a triangle ABC, a = 700 feet, B = 73° 48', and C = 37° 21'. If M is the middle point of BC find the length of AM, and the angles BAM and MAC.
3. Three circles of radii 3, 4, and 5 touch each other externally. Find the angles of the triangle formed by joining their centers.
4. A and B are points on opposite sides of a river. On one bank the line AC 650 feet is measured. The angle A = 73° 40', and C = 52° 38'. Find AB.
5. In a cube, the
diagonal is
15 ft. Find the area and volume of the cube.
6. The sides of
a parallelogram are
AB = 209.16 and AD
=347.25, and the diagonal AC = 351.47. Find the angles
and the other diagonal.
7. In a parallelogram ABCD, the diagonal AC = 521.16, then angle ABC = 110° 48' 12", and BAC = 27° 19' 36". Find the lengths of the sides and the other diagonal.
8. The diagonals of a parallelogram are 374.14 and 427.21 and the included angle is 70° 12' 38". Find the sides.
9. The sides of a quadrilateral in order
are 763.83, 721.75,
547.12, and 593.21, and the angle between the first
two sides is 53°
13' 12". Find the other three angles.
10. In a cuboid, the diagonal is 25 ft and two of its sides are 10
and 20. What is the volume of the cuboid?
11.
On one side of
a stream lines
PA = 586.3 feet, PB = 751.6
feet are measures, angle APB being 167°
36'. Q is a point on the opposite
side of the stream. Angle PAQ = 63°
18' and PBQ = 49° 24'. Find
PQ.
12. In the pyramid VABCD (with ABCD the base), we have: AC = 10 ft, DCA = 40°, VCA = 65°. Find the total area and the volume of the pyramid.
1. You can use the law of sines to determine either of the lengths AB or BC. The question is to find the distance from C to AB. That means you drop a perpendicular from C to that line and determine its length. You could use the angle A and the line AC to find it, or you could use the angle B and the line BC to find it.
2. Same hint as 1.
3. The circles are tangent, so a line from one center to another is the sum of the radius of one circle and that of the other. You've got a triangle with sides 7, 8, and 9. You can use the law of cosines to find the angles.
4. The law of sines works well here.
6. You know the sides of triangles ABC and ADC, so you can determine their angles. In triangle ABD you then know an angle and the two adjacent sides, so you can find the opposite side BD.
7. First solve the triangle ABC. Next in triangle ABD you know two sides and you can easily determine the angle BAD.
8. The "included angle" is one of the two angles between the two diagonals. The other included angle is its supplement 180° ? 70° 12' 38". Let P be the point where the two diagonals meet. It is the midpoint of each diagonal, so you know the distance between P and any vertex. Use the law of cosines to on two triangles with vertices P and two of the vertices of the parallelogram.
9. You know the sides of the quadrilateral ABCD and the angle at B. You can solve triangle ABC. Then you know all the sides of triangle ACD, so you can find its angles.
11. First solve the triangle APB. Then you'll have enough information to solve the triangle AQB.
1. 345.43 feet.
2. 490.83 feet.
3. 48 ° 11' 24", 58 ° 24' 42", 73° 23' 54".
4 640 feet 10 inches.
6. 106° 18' 46", 73° 41' 14", 452.92.
7. 255.93, 372.11, 369.22.
8. 231.94, 328.93.
9. 125° 6' 12", 70° 57' 54", 110° 42' 42".
11. 854.6 feet.