Chapter 8: Right Triangles Topic 5: Mean Proportions & Altitude Rules
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1 Name: Date: Do Now: Use the diagram to complete all parts: a) Find all three angles in each triangle. Chapter 8: Right Triangles Topic 5: Mean Proportions & Altitude Rules b) Find side ZY c) Are these triangles similar? Explain your answer. Exploring a Proportion - Remember the rule is that the product of the means equals the product of the extremes. - This proportion can be referred to as Mean Proportional or Geometric Mean Examples: 1) Find the geometric mean of 4 and 18 in simplest 2) 25 is the geometric mean of 5 and what number? radical form. 3) is the geometric mean of 6 and what number?
2 Overlapping Triangles Notes The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. Since these triangles are similar, we can establish proportions relating the corresponding sides and solve for missing pieces of information. Lets break apart the triangles: Large Triangle Left Triangle Right Triangle The large and left triangles both contain: Now the left triangle is rotated clockwise. 1) a right angle <ACB and <ADC The left and right both contain: 2) <A (Reflexive Property) 1) right angles <ADC and <CDB 2) <CAD rotated to <BCD Example: 1) Find the value of x. Steps: Separate the triangles Set up a proportion to solve Practice: 2) Find the value of x. Can you identify a pattern?? ALTITUDE RULE:
3 Practice: 5) Find the value of x. 6) Find the value of x. 7) Find the value of x. 8) Find the value of x
4 Name: Date: Mean Proportions and Overlapping Triangles Homework 1) Find the geometric mean of 3 and 48. 2) Find the geometric mean of 4 and 10 in simplest radical form. 3) is the geometric mean of 12 and 4) Find the value of x. what number? 5) Find the value of x. 6) Find the value of x. 7) Find the value of x.
5 Review Section: 8.) In m<a = 95 and m<b = 50 and m<c = 35. Which expression correctly relates the lengths of the sides of this triangle? (1) (2) (3) (4) 9.) Point P is on line m. What is the total number of planes that are perpendicular to line m and pass through point P. (1) 1 (2) 2 (3)0 (4) infinite 10.) 7, 9, and 10 can be the lengths of the sides of a triangle. (1) true (2) false 11.) In the diagram below of trapezoid RSUT, RS TU, X is the midpoint of RT, and V is the midpoint of SU. If RS=3x + 7, XV=3x + 17, and TU = 5x + 11, find the value of x and the length of all three segments. 12.) The vertex of an isosceles triangle is four times the measure of a base angle. Find the measure of all three angles of the isosceles triangle. 13.) In two complementary angles, the measure of one angle is 6 more than twice the measure of the other. Find the measure of both angles.
6 Name: Do Now: Find the value of x. Chapter 8: Right Triangles Topic 5: Mean Proportions & Right Triangle Rules DAY 2 Date: 1) Find the value of x. 2) Find the value of x. 3) Find the value of x. 4) Find the value of x. 5) Find the value of x.
7 6) Find the value of x and y in simplest radical form. 7) Find the value of x, y and z to the nearest tenth. 8) Find the value of x. 9) Find the value of x, y and z to the nearest hundredth.
8 Name: Date: Altitude & Leg Rule Day 2 Homework 1) Find the value of x. 2) Find the value of x. 3) Find the value of x. 4) Find the value of x, y and z to the nearest tenth.
9 5) Find the value of x. Review Section: 6) Express in simplest radical form. 7) In and PM is the shortest side of, what is the shortest side of (1) XV (2) WX (3) VW (4) NP 8) As shown in the diagram below, CD is the median of. Which statement is always true? (1) (2) (3) (4) 9) Plane A and Plane B are two distinct planes that are both perpendicular to line l. Which statement about planes A and B is true? (1) Planes A and B have a common edge, which forms a line (2) Planes A and B are peperndicular to each other (3) Planes A and B intersect each other at exactly one point (4) Planes A and B are parallel to each other 10) ABC is similar to DEF. The lengths of the sides of are 5, 8, and 11. What is the length of the shortest side of if its perimeter is 60? 11) The equation of a line is. What is an equation of the line that is perpendicular to the given line and that passes through the point (4,2)?
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