angle of depression angle of elevation Angle of Depression: the angle between the horizontal and the line of sight to an object below the horizontal. Angle of Elevation: the angle between the horizontal and the line of sight to an object above the horizontal. angle of depression = angle of elevation
(1). A forest ranger is on a fire lookout tower in a national forest. His observation position is 214.7 feet above the ground when he spots an illegal campfire. The angle of depression of the line of site to the campfire is 12°. a. The angle of depression is equal to the corresponding angle of elevation. Why? b. Assuming that the ground is level, how far is it from the base of the tower to the campfire?
(1a). The angle of depression is equal to the corresponding angle of elevation. Why? (1b). Assuming that the ground is level, how far is it from the base of the tower to the campfire? Alternate Interior Angles are Congruent 214.7 x The fire is 1,010.1 feet from the base of the tower.
(2). A ladder 5 m long, leaning against a vertical wall makes an angle of 65˚ with the ground. a) How high on the wall does the ladder reach? b) How far is the foot of the ladder from the wall? c) What angle does the ladder make with the wall? 5 m x The ladder reaches 4.5 m up the wall. 65o
(2). A ladder 5 m long, leaning against a vertical wall makes an angle of 65˚ with the ground. b) How far is the foot of the ladder from the wall? 5 m 65o The foot of the ladder is 2.1 m from the wall. y
(2). A ladder 5 m long, leaning against a vertical wall makes an angle of 65˚ with the ground. c) What angle does the ladder make with the wall? zo 5 m The ladder makes a 25o angle with the wall. 65o
(3). The ends of a hay trough for feeding livestock have the shape of congruent isosceles trapezoids as shown in the figure below. The trough is 18 inches deep, its base is 30 inches wide, and the sides make an angle of 118° with the base. How much wider is the opening across the top of the trough than the base? x 118o – 90o = 28o 18 28o 90o The opening across the top is 2(9.6) or 19.2 in. wider than the base.
Investigation 1: In your group, select one person as the recorder, one as the calculation expert, and one as the presenter. Work the problem assigned to your group on poster paper including a labeled diagram, equation with solution, and answer written in sentence form. The presenter will demonstrate your solution to the class.
(4). A guy wire is attached from the top of a tower to a point 80m from the base of the tower. (a). If the angle of elevation to the top of the tower of the wire is 28°, how long is the guy wire? (b). How tall is the tower? (4a). (4b). y x The wire is 90.6 m long and the tower is 42.5 m tall. 28o 80 m
(5). How tall is a bridge if a 6-foot-tall person standing 100 feet away can see the top of the bridge at an angle of elevation of 30° to the horizon? x The bridge is 57.7 + 6 or 63.7 ft tall 30o 100 ft 6 ft 100 ft
(6). A bow hunter is perched in a tree 15 feet off the ground. The angle of depression of the line of site to his prey on the ground is 30o. How far will the arrow have to travel to hit his target? 30o 15 x The arrow will travel 30 ft to its target. 30o
(7). Standing across the street 50 feet from a building, the angle to the top of the building is 40°. An antenna sits on the front edge of the roof of the building. The angle to the top of the antenna is 52°. a) How tall is the building? x The building is 42 ft tall. 40o 50 ft
(7). Standing across the street 50 feet from a building, the angle to the top of the building is 40°. An antenna sits on the front edge of the roof of the building. The angle to the top of the antenna is 52°. b) How tall is the antenna itself, not including the height of the building? y The antenna is 64.0 – 42.0 or 22 feet tall. 52o 50 ft
(8). An air force pilot must descent 1500 feet over a distance of 9000 feet to land smoothly on an aircraft carrier. What is the plane’s angle of descent? xo 1500 ft xo 9000 ft The angle of descent is approximately 9.5o.
Homework #1-7at bottom of Note Handout 1.5 Applications of Right Triangles
Intro to Circles and Properties of Tangents Section 9.1 Standard: MCC9-12.G.C.1 Essential Question: How are circles tangents used to solve problems?
D C B A F E Introduction:A circle is the set of all points in a plane at a given distance from a given point. A. A circle is named by its center. The circle shown below has center C so it is called circle C. This is symbolized by writing C.
D C B A F E B. Draw a line segment by connecting points C and D in the circle above. The segment you have drawn is called aradius.The plural of radius is radii. Name three other radii of the circle. Be sure to use the correct notation for a line segment: ______ , ______ , ______ Note: one endpoint of the radius is the center of the circle and the other endpoint is a point on the circle. Also, all radii of a circle are congruent. So, if AC = 2 cm, then CE = _____ cm.
D C B A F E C. Now, make another segment by connecting points A and E. The segment you have drawn is called a chord. Name five other chords of the circle. ______ , ______ , ______ , ______ , ______ Note: both endpoints of a chord are points on the circle. Chords of a circle do not necessarily have the same length.
D C B A F E D. If a chord passes through the center of a circle it is given a special name. It is called a diameter. Name the diameter picturedin C. _________ Now, draw that diameter. Note: a diameter is the longest chord of a circle. Its length is twice that of the radius. So, if AC = 2 cm, then AB = _____ cm.
D C B A F E E. Next, draw a line passing through points B and F. The line you have drawn is called atangent. A tangent lies in the plane of the circle and intersects the circle in only one point. The point of intersection is called the point of tangency. What is the point of tangency for BF? ________
D C B A F E F. Now, draw the line passing through points A and F. This line is called a secant. Any line that contains a chord of a circle is called a secant.
Use the circle below to identify the following: • Name the circle ________ • Name all radii pictured. • _____________ • Name all of chords pictured. • __________ • Name a diameter. • ________ • Name a tangent. • ________ • Name a secant • ________ K X Y K Z W
The points located inside a circle are called interior points. The points located on the circle are points of the circle itself. The points located outside the circle are called exterior points. Draw four lines that are tangent to both of the circles below at the same time. These lines are called common tangents.
The two common tangents that pass between the two circles above are called internal common tangents. The other two tangents are called external common tangents.
Draw all the common tangents possible for the problems below. 1. 2. 3.
Summary: Circle: The set of all points in a plane at a given distance from a given point is a circle. P is the set of all points in the plane that are 2 units from P. The given point P is the center of the circle. Radius: The given distance is the radius of the circle. A radius is also a segment joining the center of the circle to a point of the circle. (The plural of radius is radii.) Chord: A segment whose endpoints lie on a circle is a chord. Diameter: A chord that contains the center of a circle is a diameter. A diameter is also the length equal to twice a radius. P
Secant: A line that contains a chord of a circle is a secant. Tangent: A line in the plane of a circle that intersects the circle in exactly one point is a tangent. The point of tangency is the point of intersection. secant diameter radius chord tangent
REMEMBER: a tangentis a line in the plane of a circle that intersects the circle in exactly one point, the point of tangency. A tangent rayand a tangent segment are also called tangents.
Theorem 1: In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle (the point of tangency). For the figure at right, identify the center of the circle as O and the point of tangency as P. Mark a square corner to indicate that the tangent line is perpendicular to the radius. P O
Theorem 2 : Tangent segments from a common external point are congruent. Measure and with a straightedge to the nearest tenth of a cm. RS = _______ cm RT = ______ cm 2.6 2.6 S 2.6 cm R 2.6 cm T
Example 1: In the diagram below, is a radius of circle R. If TR = 26 , is tangent to circle R? Right Triangle? 102 + 242 = 262 100 + 576 = 676 676 = 676 Therefore, ∆RST is a right triangle. So, is tangent to R. R 26 T 10 24 S
Example 2: is tangent to C at R and is tangent to C at S. Find the value of x. 32 = 3x + 5 27 = 3x 9 = x R 32 Q 3x + 5 S
Example 3: Find the value(s) of x: x2 x2 = 16 x = ±4 Q R 16 S
Example 7: Use the diagram at right to find each of the following: 1. Find the length of the radius of A. 2. Find the slope of the tangent line, t. D = ≈ 2.8 m = t A (3, 1) Radius and tangent are perpendicular, hence their slopes are opposite reciprocals (5, -1)
Properties of Chords Section 9.2 Standard: MCC9-12.G.C.2 Essential Question: Can I understand and use properties of chords to solve problems?
Parts of circles are called Arcs. If the part of the circle is less than half the circle it is called a minor arc. If the part of the circle is more than half the circle it is called a major arc. And if it is exactlyhalf the circle it is called a semicircle.
The circle below has center P. Draw a diameter on your circle. Label the endpoints of the diameter H and K. Put another point on your circle and label it T. There are two semicircles pictured in your drawing. One can be symbolized HK and the other HTK . K T P H
K T P Name two minor arcs(you only need two letters to name a minor arc because you always travel the shortest distance unless told otherwise.) _______ , _______ Name two major arcs(you must use three letters to name a major arc because you always travel the shortest distance unless told otherwise.) _______ , _______ H HT TK KHT HKT
The circle below has center M. Mark two points on your circle and label them R and S. Now draw the following rays: and Color the inside of RMS. Name the arc that is inside the colored part of the angle. This is called the intercepted arc. Arc: ________ The angle is called acentral anglebecause its vertex is the center of the circle. S R M RS
S R M Arcs are measured in degrees. The measure of the intercepted arc is equal to the measure of the central angle that forms that arc. So, if mRMS = 60o, then m RS = 60o.
In this section you will learn to use relationships of arcs and chords in a circle. In the same circle, or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. C B AB DC if and only if _____ ______. CD AB D A
1. In the diagram, A D, , and m EF = 125o. Find m BC. E B A D F C chords Because BC and EF are congruent ________ in congruent _______, the corresponding minor arcs BC and EF are __________ . So, m ______ = m ______ = ______o. circles congruent 125 BC EF
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. If QS is a perpendicular bisector of TR, then ____ is a diameter of the circle. T S P Q R QS
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. If EG is a diameter and EG DF, then HD HF and ____ ____ . F E H G D GF GD
2. If m TV = 121o, find m RS. T 6 S V 6 R If the chords are congruent, then the arcs are congruent. So, m RS = 121o
3. Find the measure of CB, BE, and CE. C Since BD is a diameter, it bisects, the chord and the arcs. 4xo A B D 4x = 80 – x 5x = 80 x = 16 (80 – x)o E 4(16) = 64 so mCB = m BE = 64o mCE = 2(64o) = 128o
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. if and only if _____ ____ C GE FE G E D A F B
4. In the diagram of F, AB = CD = 12. Find EF. Chords and are congruent, so they are equidistant from F. Therefore EF = 6 G A B 7x – 8 F 3x D E C 7x – 8 = 3x 4x = 8 x = 2 So, EF = 3x = 3(2) = 6
In the diagram of F, suppose AB = 27 and • EF = GF = 7. Find CD. Since and are both 7 units from the center, they are congurent. G A B F D E C So, AB = CD = 27.