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Solving 7th-grade Math on pages 81 and 82 of Volume 2 of the Creative Horizon book: The properties of the three angle bisectors of a triangle are valuable reference materials. With detailed solutions to exercises 1, 2, 3... on pages 81, 82 according to the textbook curriculum, students can quickly tackle exercises and master the knowledge of angle bisectors in a triangle.
Other excellent study materials for Grade 7 Mathematics:
- Solving 7th-grade Math with Creative Horizon book
- Solving 7th-grade Math on page 76 of Volume 2 of the Knowledge Connection - Exercise 34: The concurrency of three medians, three angle bisectors in a triangle
- Solving 7th-grade Math on page 111 of Volume 2 of the Kite book - Exercise 11. Properties of the three angle bisectors of a triangle
Solving 7th-grade Math on pages 81 and 82 of the textbook Volume 2, Creative Horizon book
Properties of the three angle bisectors of a triangle
1. Solve Exercise 1 Page 81 Mathematics Textbook Grade 7
Problem: In Figure 8, I is the intersection of the three angle bisectors of triangle ABC.
a) Given IM = 6 (Figure 8a). Find IK and IN.
b) Given IN = x + 3, IM = 2x - 3 (Figure 8b). Find x.
Solution: The three angle bisectors of a triangle intersect at a common point. This point is equidistant from the three sides of the triangle.
Answer:
a) Since I is the intersection of the three angle bisectors of triangle ABC, I is equidistant from the three sides of triangle ABC.
Therefore, IM = 6. Consequently, IN = IK = 6.
b) Since IM = IN, then 2x - 3 = x + 3.
2. Solve Exercise 2 Page 82 Mathematics Textbook Grade 7
Problem: For isosceles triangle ABC at A. Draw median AM. Angle bisector of angle B intersects AM at I. Prove that CI is the angle bisector of angle C.
Solution Guide:
Utilize the property: The three angle bisectors of a triangle intersect at a common point.
Answer:
As triangle ABC is isosceles at A, AM is the median of the triangle, so AM is also the angle bisector of angle A.
The angle bisector of angle B intersects AM at I, making I the intersection of the three angle bisectors in triangle ABC. Therefore, CI is also the angle bisector of the triangle.
Therefore, CI is the angle bisector of angle C.
3. Solve Exercise 3 Page 82 Mathematics Textbook Grade 7
Problem: For isosceles triangle ABC at A. Angle bisectors of angles B and C intersect at M. Line AM intersects BC at H. Prove that H is the midpoint of BC.
Solution Guide:
In an isosceles triangle, the angle bisector originates from the vertex opposite the base and simultaneously serves as the median corresponding to the base.
Answer:
As the angle bisectors of angles B and C intersect at M in triangle ABC, M is the intersection of the three angle bisectors in triangle ABC. Therefore, AM is the angle bisector of angle A.
Line AM intersects BC at H. Therefore, AH is the angle bisector originating from vertex A.
As triangle ABC is isosceles at A, AH is the median corresponding to base BC. Therefore, H is the midpoint of BC.
4. Solve Exercise 4 Page 82 Mathematics Textbook Grade 7
Problem: For triangle DEF. Angle bisectors of angles D and E intersect at I. Through I, draw a line parallel to EF, this line intersects DE at M, and intersects DF at N. Prove that ME + NF = MN.
Solution Guide:
Utilize the property: The three angle bisectors of a triangle intersect at a common point.
Prove: ME = MI, NF = NI.
Answer:
5. Solve Exercise 5 Page 82 Mathematics Textbook Grade 7
Problem: For right-angled triangle AMN at A. The angle bisectors of angles M and N intersect at I. Line MI intersects AN at R. Draw a perpendicular from I to AI and label the intersection point as T. Prove that AT = RT.
Solution Guide:
Utilize the property: The three angle bisectors intersect at a common point.
In a right-angled triangle, if one angle is 45 degrees, then the triangle is a right-angled isosceles triangle.
Answer:
6. Solve Exercise 6 Page 82 Mathematics Textbook Grade 7
Problem: Three cities A, B, C are connected by highways (Figure 9). One wants to find a location to build an airport so that it is equidistant from all three highways. Determine the position of the airport satisfying the given condition and explain the procedure.
Solution Guide: The three angle bisectors of a triangle intersect at a common point. This point is equidistant from the three sides of the triangle.
Answer:
Cities A, B, C are connected by highways forming triangle ABC. Let I be the airport location. Since the airport must be equidistant from all three highways, I is equidistant from the three sides of triangle ABC. Thus, I is the intersection of the three angle bisectors in triangle ABC.
To determine I, draw two angle bisectors of angle B and angle C. I is the intersection of these two angle bisectors.
Here is the guide to solve Grade 7 Mathematics on pages 81 and 82, students can also refer to the solution to Grade 7 Mathematics on page 84 and review Grade 7 Mathematics on page 78 to solidify their knowledge.
- Grade 7 Mathematics on page 84 of Creative Horizon book - Exercise at the end of Chapter 8
- Grade 7 Mathematics on page 78 of Creative Horizon book - Exercise 8: Properties of the three altitudes of a triangle
