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Detailed solutions and solving guides for exercises 4.23 to 4.28 of Isosceles triangle, perpendicular bisector of the segment are all included in the document Solving 7th grade math problems on page 84, book Connecting Knowledge with Life. Students can refer to these to excel in Mathematics and thus improve their grades.
Referencing multiple learning materials for 7th-grade Mathematics:
- Solving 7th-grade Mathematics problems book Connecting Knowledge with Life
- Solving 7th-grade Mathematics problems on pages 62, 63, Volume 2 of Creative Horizon - Lesson 3: Isosceles triangle
- Solving 7th-grade math problems on page 96, Volume 2 of Kite - Lesson 7. Isosceles triangle
Solving 7th Grade Math Problems on Page 84, Book: Connecting Knowledge with Life
Isosceles Triangle. Perpendicular Bisector of the Segment
1. Solve Exercise 4.23 Page 84 Math Textbook Grade 7
Problem Statement: Given isosceles triangle ABC with A as the apex and points E, F lying on sides AC, AB respectively such that BE is perpendicular to AC, CF is perpendicular to AB (H.4.69). Prove that BE = CF.
Solution Guide:
Prove two triangles congruent to deduce two corresponding sides equal
Answer:
2. Solve Exercise 4.24 Page 84 Math Textbook Grade 7
Problem Statement: Given isosceles triangle ABC with A as the apex and M as the midpoint of segment BC. Prove that AM is perpendicular to BC and AM is the angle bisector of angle BAC.
Guide to solve: Prove two triangles AMC and AMB congruent to deduce corresponding pairs of angles equal.
Answer:
Considering triangles AMC and AMB:
AM is common
AB equals AC (due to triangle ABC being isosceles at A)
MB equals MC
3. Solve Exercise 4.25 Page 84 Math Textbook Grade 7
Problem Statement: Given triangle ABC and M as the midpoint of segment BC.
a) Assuming AM is perpendicular to BC. Prove that triangle ABC is isosceles at A.
b) Assuming AM is the angle bisector of angle BAC. Prove that triangle ABC is isosceles at A.
Solution Guide:
a) Prove that triangles AMB and AMC are congruent. Then, identify a pair of corresponding sides or angles to infer that triangle ABC is isosceles.
b) From point M, draw two lines perpendicular to AC and AB respectively, then prove that angles B and C are equal.
Answer:
a)
b)
Draw MH perpendicular to AB (H lies on AB)
Draw MK perpendicular to AC (K lies on AC)
4. Solve Exercise 4.26 Page 84 Math Textbook Grade 7
Problem Statement: A right triangle with two equal sides is called an isosceles right triangle.
Explain the following statements:
a) In an isosceles right triangle, the vertex angle is at the right angle;
b) In an isosceles right triangle, both acute angles are 45°;
c) A right triangle with one acute angle of 45° is an isosceles right triangle.
Apply the sum of the three angles in a triangle equals 180 degrees.
Answer:
5. Solve Exercise 4.27 Page 84 Math Textbook Grade 7
Problem Statement: In Figure 4.70, which line is the perpendicular bisector of segment AB?
A line perpendicular to a segment at its midpoint is called the perpendicular bisector of that segment.
Answer:
Observing figure 4.70, we see that line m is perpendicular to segment AB at its midpoint, so m is the perpendicular bisector of AB.
6. Solve Exercise 4.28 Page 84 Math Textbook Grade 7
Problem Statement: Given isosceles triangle ABC with A as the apex and altitude AD. Prove that line AD is the perpendicular bisector of segment BC.
Solution Guide:
Prove two congruent triangles to infer two corresponding sides are equal, two corresponding angles are equal
Note: Two adjacent supplementary angles are each 90 degrees
Answer:
Here is the guide to solve 7th-grade math problems on page 84. Students can refer to it before moving on to page 86 and reviewing page 79 for a better understanding.
