Unit 5 Similar Polygons Lesson 15-22

Lesson 15 Using Triangles- Rectangular Solids

A rectangular solid is a figure formed by six rectangles. The six rectangles are called its faces. The opposite faces are parallel and the adjacent faces are perpendicular.

If we take rectangle ABCD to be the base, any face that intersects the base is called a lateral face. ABFE, BCGF, CDHG,and ADHE are lateral faces.An edge is a segment formed by the intersection of two faces.

For example,  ,  , and   are a few of the sides of the rectangular solid.

A vertex is any point where two edges intersect. A, B, C, D, E, F, G, and H are the vertices.The diagonal of a face is a segment whose endpoints are nonconsecutive vertices in the same face.

The diagonal of the rectangular solid is a segment whose endpoints are vertices not in

the same  face.  

A cube is a special rectangular solid where all the faces are squares. Since all the faces are squares, all the edges are equal. Using the formula for the diagonal of a rectangular solid where l = w = h = e, 

The diagonal of a cube is the length of the edge times 

Lesson 16 Using Triangles- Regular Square Pyramid

A regular square pyramid has a base that is a square and lateral faces that are congruent isosceles triangles. The altitude of the pyramid is a segment from the vertex perpendicular to the base. PQ is the altitude of the pyramid.The slant height  is the altitude of one of the lateral faces. PR is the slant height of the pyramid.

Right triangles are formed using the altitude, slant height, and edge. Again we can use the Pythagorean formula to find missing dimensions.

Model: Find l and k for this pyramid where AB = 6, QR = 3 , and PQ = 4.

Solution: In triangle PQR:

l 2 = (PQ)2 + (QR)2

l 2 = 42 + 32

l 2 = 16 + 9l 2 = 25l = 5

In triangle PRC:k2 = l 2 + (RC)2

k2 = 52 + 32

k2 = 25 + 9k2 = 34

Lesson 17 Trigonometry- Sine Ratio

The word trigonometry comes from the Greek words that mean triangle measurement . 

sine ratioWith respect to an acute angle of a right triangle, the ratio formed by the side opposite the acute angle over the hypotenuse.

Take several similar right triangles and overlap them so that the acute angles at A match up. Since the triangles are all similar, the sides are in proportion. A special ratio that is formed by taking the side opposite  A over the hypotenuse is called the

sine ratio with respect to angle A. This measure is written:

 or   or   or   

or 

Since these triangles are similar, we know that   As long as we keep the same acute angle at A, the sin   A is the same number whatever right triangle we use.

Mathematicians have found the approximate values of the sine ratio for all acute angles. These values are put into a table in decimal notation along with the cosine and tangent ratio numbers. Of course, many calculators today perform this role also.

TABLE OF TRIGONOMETRIC RATIOS

Notice that as the angle gets larger, so does the sine ratio number. The sine ratio number will never be larger than 1. In a right triangle, the hypotenuse is the longest side and any fraction (ratio) with the denominator larger than the numerator will be less than 1.

Lesson 18 Trigonometry- Cosine Ratio

cosine ratio With respect to an acute angle of a right triangle, the ratio of the side adjacent to the angle over the hypotenuse.

Another ratio that can be made with the similar right triangles is the side adjacent to the acute angle over the hypotenuse.

This ratio is called the cosine ratio with respect to   A. This measure is

written: 

or

In the figure above the ratios are all equal; therefore, the cosine of angle A is the same number, whichever triangle we use.

In the models below angle A is the value being illustrated.

Model 1:  

 

model 3:  

 

The cosine number will always be less than 1. Also notice that the sine of an angle and the cosine of the complement of that angle are the same number.

Models:sin 30° = cos 60°sin 60° = cos 30°sin 40° = cos 50°sin 45° = cos 45°

Lesson 19 Trigonometry- Tangent Ratio

tangent ratio With respect to an acute angle of a right triangle, the ratio of the side opposite the acute angle over the side adjacent to the acute angle.

The third ratio that we will use is formed by taking the side opposite angle A over the side adjacent to angle A.

This ratio is called the tangent ratio with respect to angle A. We write

 

Just as with the sine and the cosine, the tangent ratio is the same for a given angle no matter how large the sides of the triangle become. For tangent ratio. Unlike the sine and cosine ratio numbers, the tangent ratio number can be greater than one. As the angle increases so does the tangent number without a limit.

For special triangles such as 30-60-90 and 45-45-90 the tangent ratio can be found without the use of the tables.

 

 

Lesson 20 Using Similar Triangles in Indirect Measurement

At the bank of a river, by measurement, a = 9 ft., b = 15 ft., c = 12 ft., and d = 7 ft.How long is x?

A boy is 6 ft. tall. The distance from the boy to a mirror is 8 ft. From the mirror to the house is 16 ft. How high is the top of the house?

3 ft.

12 ft.

21  ft.

 WORK THIS CHALLENGE

EXERCISE. 

A is known to be 6,500 feet above sea level; AB = 600 feet. The angle at A looking up at P is 20°. The angle at B looking up at P is 35°. How far above sea level is the peak P?

Find the height of the mountain peak to the nearest foot.

Height above sea level =  a0ft.

Lesson 21 Using Trigonometry in Indirect Measure