كتاب الطالب Integrated II الرياضيات للصف التاسع منهج انجليزي Reveal الفصل الثالث 2021 2022
Apply Example 3 Find a Centroid on the Coordinate Plane
CHIMES Lashaya needs to hang a wind chime with a single piece of cord. The pipes of the wind chime are attached to a triangular platform. When the platform is placed on a coordinate plane, the vertices of the triangle are located at (1, 1), (11, 5), and (7, 10). What are the coordinates of the point where the cord should be attached to the platform so the wind chime stays balanced
1 What is the task
Describe the task in your own words. Then list any questions that you may have. How can you find answers to your questions
Sample answer: I need to find the balancing point of the triangular platform. The balancing point of a triangular region is the centroid, so I need to find the centroid of the triangle that is described
2 How will you approach the task ? What have you learned that you can use to help you complete the task ?
Sample answer: I will graph the triangle on the coordinate plane. Then, I will use the Midpoint Formula to calculate the midpoint of one side of the triangle. Then, I will use the Centroid Theorem and what I have learned about calculating fractional distance to find a point that iss of the distance from the vertex opposite the midpoint that I found to the midpoint
3 What is your solution
use your strategy to solve the problem Graph the triangular platform and the medians of AABC
Explore Applying Indirect Reasoning
Online Activity Use the video to complete the Explore
INQUIRY How can you use a contradiction to prove a conclusion
Learn Indirect Proof
A direct proof is one that starts with a true hypothesis. and the conclusion is proved to be true. Indirect reasoning eliminates all possible conclusions but one, so the one remaining conclusion must true. In an indirect or by contradiction. one assumes that the statement to be proved is false and then uses logical reasoning to deduce that a statement contradicts a postulate, theorem, or one of the assumptions Once a contradiction is obtained, one concludes that the statement assumed false must in fact true
Key Concept • How to Write an Indirect
Step 1 Identify the conclusion that you are asked to prove. Make the assumption that this conclusion is false by assuming that the negation is true
Step 2 Use logical reasoning to show that this assumption leads to a contradiction of the hypothesis or some other fact such as a definition, postulate. theorem, or corollary
Step 3 State that because the assumption leads to a contradiction. the original conclusion, what you were asked to prove, must be true
In indirect proofs, you should assume that the conclusion you are trying to prove is false. If, in the proof, you prove that the hypothesis is then false, this is a proof by contrapositive. If, in the proof, you assume
that the hypothesis is true and prove that some other known fact is false, this is a proof by contradiction
Example 1 Write an Indirect Algebraic Proof
Explore Analyzing Inequalities in Two Triangles
Online Activity use dynamic geometry software to complete the Explore
INQUIRY How do the included angle measures of two triangles with two pairs of congruent sides compare
Learn Hinge Theorem
Theorem 1.12: Hinge Theorem
If two sides of a triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second triangle. then the third side of the first triangle is longer than the third side of the second triangle
You will prove Theorem 1.12 in Exercise 18
Example 1 Use the Hinge Theorem
BOATING Two families set sail on their boats from the same dock. The Nguyens sail 3.5 nautical miles north. tum 850 east of north, and then sail 2 nautical miles. The Griffins sail 3.5 nautical miles south, turn 95• east of south, and then sail 2 nautical miles. At this point, which boat is farther from the dock ? Explain your reasoning
Step 1 Draw a diagram of the situation The courses of each boat and the straight-line distance from each stopping point back to the boat dock form two triangles. Each boat sails 3.5 nautical miles. turns. and then sails another 2 nautical miles
Step 2 Determine the interior angle
HARBOR measures
Use linear pairs to find the measures of the included angles. The measure of the included angle for the Nguyens is 180 — 85 or 95'. The measure of the included angle for the Griffins is 180 - 95 or 85
Step 3 Compare the distance each boat is from the boat launch
Use the Hinge Theorem to compare the distance each boat is from the boat launch
Module Summary
Lessons 1 -1 and 1- 2
Perpendicular Bisectors and Angle
Bisectors
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment
The perpendicular bisectors of a triangle intersect at the circumcenter that is equidistant from the vertices of the triangle
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle
The angle bisectors of a triangle intersect at the incenter, which is equidistant from the sides of the triangle
Lesson 1- 3
Medians and Altitudes
A median of a triangle is a line segment with endpoints that are a vertex of the triangle and the midpoint of the side opposite the vertex
The medians of a triangle intersect at the centroid, which is two - thirds of the distance from each vertex to the midpoint of the opposite side
An altitude of a triangle is a segment from a vertex of the triangle to the line that contains the opposite side and is perpendicular to that side
The altitudes of a triangle intersect at a point called the orthocenter