Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Trigonometric functions wikipedia , lookup

History of geometry wikipedia , lookup

Multilateration wikipedia , lookup

Pythagorean theorem wikipedia , lookup

Integer triangle wikipedia , lookup

Rational trigonometry wikipedia , lookup

Cross product wikipedia , lookup

Geometrization conjecture wikipedia , lookup

Transcript

Geometry Notes G.7 Triangle Ratios, Proportions, Geometric Mean Mrs. Grieser Name: __________________________________________ Date: ________________ Block: _______ Review: Ratio the comparison of two numbers using division, written “a to b”, a:b, or Proportion an equation that states two ratios are equal; ex: Cross Products a (b ≠ 0) b a c , b 0, d 0 b d the product of the means or extremes (see diagram below); cross products are = Ratios should be expressed in simplest form, and can be used to find missing dimensions Extended ratios compare more than two numbers; e.g. ratio of sides of a triangle are 4:5:8. o Example: Find the sides of a ∆ with the above ratio if the perimeter is 204. Proportions are made up of parts called means and extremes: The Cross Product Property may be used to solve proportions: Cross Product Property In a proportion, the product of the extremes equals the product of the means: a c If , where b ≠ 0 and d ≠ 0, then ad = bc b d Examples: a) The perimeter of a rectangle is 484. The ratio of length to width is 9:2. Find the area of the rectangle. Solution: Perimeter = __________ c) Solve: 5 x 10 16 b) The measures of the angles of a triangle are in the ratio of 1:2:3. Find the measures of the angles. Solution: Sketch ∆ and write in angles according to given ratio. d) Solve: 1 2 y 1 3y Set up cross product: ________________ Set up cross product: ________________ Solve: ____________________________ Solve:____________________________ Geometry Notes G.7 Triangle Ratios, Proportions, Geometric Mean Mrs. Grieser Page 2 Geometric Mean The Geometric Mean (Mean Proportional) of two positive numbers a and b is the positive a x number x that satisfies: x2 ab x ab x b Examples: a) Find the geometric mean of 2 and 8. Solution: b) Find the geometric mean of 24 and 48. Solution: Other proportion properties: o Reciprocal Property: If two ratios are equal, then their reciprocals are equal: a c b d if then b d a c o If you interchange the means of a proportion, then you form another true proportion: a c a b if then b d c d o In a proportion, if you add the value of the denominator to the numerator, then you a c ab cd form another true proportion: if , then b d b d Examples: a b a ? then a) If x 5 b ? c) T or F: If b) If y 7 x 10 then x 10 y 7 x3 ? x 9 then 3 ? 3 y d) T or F: If 7 y x 10 then x 10 y 7 e) In the diagram below, RU RV . Find x. US VT f) g) In the diagram below, JM KN . ML NL h) A highway on a map is 9 inches long. The actual highway is 36 miles long. What is the scale of the map? Find JL and JM. In the diagram below, RU RV . Find x. US VT