Grade 7 Mathematics Exercise Page 83 Workbook Connection of Knowledge provides detailed solutions for exercises 9.31, 9.32, 9.33, 9.34, 9.35 in the textbook Focus Practice, presented scientifically. Students along with teachers can refer to it to complete exercises and prepare lesson plans according to the new curriculum standards.
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Solving Grade 7 Mathematics Exercise Page 83 Workbook Connection of Knowledge with real-life situations
Focus Practice
1. Solve Exercise 9.31 Page 83 Mathematics Textbook Grade 7
Problem: Prove that a triangle with its median and altitude starting from the same vertex coinciding is an isosceles triangle.
Solution: Prove that the triangle has two equal sides or two angles at the base equal.
Answer:
Suppose AD is both the altitude and the median of triangle ABC. We need to prove that triangle ABC is isosceles at A. Indeed:
AD is the median of triangle ABC, so we infer that D is the midpoint of segment BC.
Moreover, AD is the altitude of triangle ABC, so AD ⊥ BC at D.
Thus, AD is the perpendicular bisector of segment BC. Therefore, we have: AB = AC (perpendicular bisector property of a line segment).
Therefore, triangle ABC is isosceles at A.
2. Solve Exercise 9.32 Page 83 Mathematics Textbook Grade 7
Problem: Given three collinear distinct points A, B, C. Let d be the line perpendicular to AB at A. With point M on d, M different from A, draw line CM. Through B, draw the line perpendicular to line CM, intersecting d at N. Prove that line BM is perpendicular to line CN.
Solution: The altitudes of a triangle are concurrent at a point. That point is the orthocenter of the triangle.
Answer:
We have:
BN ⊥ CM.
CA ⊥ MN.
CA and BN intersect at point B. Hence, B is the circumcenter of triangle MNC.
This implies MB ⊥ CN
3. Solve Exercise 9.33 Page 83 Mathematics Textbook Grade 7
Problem: There is a circular metal sheet that needs a hole drilled at its center. How can you determine the center of the metal sheet?
Solution: Utilize the properties of the perpendicular bisectors of a triangle to find the center and radius of the circular metal sheet.
Answer:
+ Take three distinct points A, B, C on the circumference of the circular metal sheet.
+ Draw the perpendicular bisectors of segments AB and BC. These two perpendicular bisectors intersect at point D. Then D is the desired center of the circular metal sheet.
4. Solution for Exercise 9.34 on Page 83 of Grade 7 Math Textbook
Problem: Given triangle ABC. Draw ray At, the angle bisector of the angle formed by rays AB and the one opposite to AC. Prove that if the line containing ray At is parallel to line BC, then triangle ABC is isosceles at A.
Guidance: Prove that triangle ABC has two equal base angles.
Answer:
5. Solution for Exercise 9.35 on Page 83 of Grade 7 Math Textbook
Guidance:
Answer:
Here is the guidance for solving Grade 7 math exercises on page 83, volume 2. Students should also refer to the solutions for exercises on page 84 and review the exercises on page 81 for better understanding.
- Solve Exercise 9.35 on Page 84 of Grade 7 Math Textbook - Last exercise of Chapter 9
- Solve Exercise 35 on Page 81 of Grade 7 Math Textbook - Exercise 35: The concurrency of three perpendicular bisectors, three altitudes in a triangle
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