Chapter 7: right triangles and trigonometry Rachel Jeong

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Chapter 7 right triangles and trigonometry

• Rachel Jeong
• Period 5
• Laptop Honors Geometry

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7-5 angles of elevation and depression

• An angle of elevation is the angle between the
  line of sight and the horizontal when an observer
  looks upward
• An angle of depression is the angle between the
  line of sight when on observer looks downward and
  the horizontally.
• Find the angle of elevation of the sun when a
  7.6-meter flagpole casts a 18.2-meter shadow.
  Round to the nearest tenth of a degree.

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Example 2 for Angle of elevation and depression

• Example2
• ADCE, so ECD is congruent to ADC and ECB is
  congruent to ABC as they are the alternate
  interior angles. So, mADC mECD 8 and mABC
  mECB 11. In the right triangle ABC with right
  angle at A,
• Cross multiply AB Tan1160.
• Divide both sides by tan 11. AB60/tan11
• AB 308.7
• In the right triangle ADC with right angle at A
• Cross multiply AD tan860
• Divide both sides by tan 8. AD60/tan8
• AD 426.9
• Three point A, B and C are collinear and B is in
  between the points A and D.
• So, BD AD AB.

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Example 3 for Angle of elevation and depression

• Example 3
• The situation is sketched as shown in the figure.
  Kirk's eyes are at point B and the geyser reaches
  the point C. We have to find the angle of
  elevation to the top of spray. The angle of
  elevation is the angle between the line of sight
  and the horizontal when a person looks upward.
  So, here the angle of elevation is mCBD
• In the right triangle BCD with right angle at D,
  TanCBDCD/BD
• We know CE and AE. We have to find CD and BD.
• Because AE and BD are horizontal lines, AEBD.
  Also the lines AB, DE are perpendicular to the
  horizontal lines. So, the quadrilateral formed is
  a rectangle. We have DE AB 6 and BD AE
  200.
• Three points C,D and E are on a straight line and
  so we have CD CE DE.
• CD 175 6 169

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7-3 special right triangles

• In 45-45-90 triangle, the length of the
  hypotenuse is v2 times the length of a leg
• In a 30-60-90 triangle, the length of the
  hypotenuse is twice the length of the shorter
  leg, and the length of the longer leg is v3 times
  the length of the shorter leg.
• Example1
• The perimeter of an equilateral triangle is three
  times the measure of a side.
• Perimeter 3x, if the measure of the side is x.
• So, 45 3x.
• Divide by 3 on both the sides.
• x 45/3 15 cm
• So, the measure of the side is 15 cm
• The altitude of an equilateral triangle divides
  the triangle into two congruent 306090
  triangles. The length of the altitude is the
  length of the longest leg in the 306090
  triangle.

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Example2 for special right triangle

• Example 2
• Given that the mDHB 60. Also the mBDH 90.
• So, the triangle BDH is a 306090 triangle.
• The side opposite to larger angle is longer leg.
  So, BDis the longer leg and DHs the shorter leg.
  From Theorem 7.7, in a 306090 triangle, the
  length of the longer leg is v 3 times the length
  of the shorter leg.
• Substitute BD
• From Theorem 7.7, in a 306090 triangle, the
  length of the hypotenuse is twice the length of
  the shorter leg.
• So, BH2(DH)
• Substitute DH
• BH2(8)
• BC16

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Example3 for special right triangle

• Example 3
• From the Theorem 7.6, the length of the
  hypotenuse is v 2 times the length of a leg in
  454590 triangle.

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7-1 Geometric means

• Geometric mean for two positive numbers a and b,
  the geometric mean is the positive number x where
  the proportion axxb is true. This proportion
  can be written using fractions as a/xx/b or with
  cross products as x²ab or xvab
• Theorem 7.1 if the altitude is drawn from the
  vertex of the right angle of a right triangle to
  its hypotenuse, then the two triangles formed are
  similar to the given triangle and to each other
• Theorem 7.2 the measure of an altitude drawn
  from the vertex of the right angle of a right
  triangle to its hypotenuse is the geometric mean
  between the measures of the two segments of the
  hypotenuse
• Theorem 7.3 if the altitude is drawn from the
  vertex of the right angle of a right triangle to
  its hypotenuse, then the measure of a leg of the
  triangle is the geometric mean between the
  measures of the hypotenuse and the segment of the
  of the hypotenuse adjacent to that leg
• Example 1 Applying Theorem 7.3 for the leg BD
• Since AC 10, BC 6 and CD x, the above
  fraction can be written as
• Find the cross products.
• 10x 6 6
• 10x 36
• Dividing by 10 on both sides
• x 36/10 3.6
• So, CD x 3.6.
• From the figure, x y 10.
• 3.6 y 10
• y 10 3.6 6.4
• We get the values as x 3.6 and y 6.4

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Example 2 for Geometric mean

• Example 2
• Given BD 5 and CD 9
• Let AD be x.
• Find the cross products.
• x x 5 9
• x2 45
• Take the positive square root of each side.
• Simplifying, we get
• x 6.7
• So, AD is about 6.7.

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Example 3 for Geometric mean

• Example 3
• Applying Theorem 7.3 for the leg BC
• Since AC15, BC5, and CDy, the above fraction
  can be written as
• Find the cross products.
• 15 y 5 5
• 15y 25
• y 25/15
• y 5/3
• Given x y 15
• Substitute the value of y as 5/3.
• x 5/3 15
• x 15 5/3
• x 40/3
• Find the cross products.