كتاب الطالب Integrated II الرياضيات للصف التاسع منهج انجليزي Reveal الفصل الثالث 2021 2022

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