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Solution for exercises 9.26, 9.27, 9.28, 9.29, 9.30 in the textbook is available on page 81, volume 2 of Connect Knowledge book. Students unsure about which problems to solve or uncertain about their solutions can benefit from consulting this helpful resource.
Explore more excellent study materials for 7th-grade Math:
- Solving 7th-grade Math Connect Knowledge book
- Solving 7th-grade Math page 70, volume 2 of Creative Horizon book - Exercise 5: The perpendicular bisector of a line segment
- Solving 7th-grade Math page 115, volume 2 of Kite book - Exercise 12: Properties of three perpendicular bisectors in a triangle
Solving 7th-grade Math page 81, volume 2, Connect Knowledge book with real-life connections
The concurrency of three perpendicular bisectors and three altitudes in a triangle
1. Solve Exercise 9.26 Page 81 Math Textbook Grade 7
Problem: Let H be the orthocenter of non-right triangle ABC. Find the orthocenter of triangles HBC, HCA, HAB.
Solution:
The altitudes of a triangle are concurrent at a point. That point is the orthocenter of the triangle.
Answer:
In triangle ABC, let H be the orthocenter. We assume AH ⊥ BC at N, BH ⊥ AC at P, CH ⊥ AB at M.
In triangle AHB, we have:
Assuming AH ⊥ BC at N, BH ⊥ AC at P, CH ⊥ AB at M.
AP is perpendicular to BH
BN is perpendicular to AH
Since AP and BN intersect at C, it follows that C is the orthocenter of triangle AHB.
In triangle HAC, we have:
HP is perpendicular to AC
CN is perpendicular to AH
Since HP and CN intersect at B, it implies that B is the orthocenter of triangle HAC.
In triangle HBC, we have:
HN is perpendicular to BC
BM is perpendicular to CH
Since HN and BM intersect at A, it implies that A is the orthocenter of triangle HBC.
2. Solve Exercise 9.27 Page 81 Math Textbook Grade 7
Solution:
In a right-angled triangle, the sum of the measures of two acute angles is 90 degrees.
Answer:
3. Solve Exercise 9.28 Page 81 Math Textbook Grade 7
Problem: Consider the scenario where point O is equidistant from the three vertices of triangle ABC. Prove that if O lies on an edge of triangle ABC, then ABC is a right-angled triangle.
Solution:
4. Solve Exercise 9.29 Page 81 Math Textbook Grade 7
Problem: a) A machine part (with a circular outer boundary) is broken (H.9.46). How can we determine the radius of this circular boundary?
b) On the map, three residential areas are planned at points A, B, C not in a straight line. Find a point M on the map equidistant from A, B, C to plan a school.
Solution:
a) Use the properties of the perpendicular bisectors of a triangle to find the center and radius of the disc.
b) Point M is equidistant from 3 points A, B, C, so M is the centroid of triangle ABC.
Answer:
a)
+ Take three distinct points A, B, C on the outer boundary of the machine part.
+ Draw perpendicular bisectors of sides AB and BC. These bisectors intersect at point O. In this case, O is the center to be determined.
+ The radius of the circle to be found is the length of segment OB (or OA or OC).
b) Since the school is located at point M equidistant from the 3 residential points A, B, C, M is the centroid of triangle ABC.
+ Draw perpendicular bisectors of segments AB, AC, BC.
+ These 3 perpendicular bisectors intersect at M. Then, MA = MB = MC.
Then, M is the point chosen for planning the school.
5. Solve Exercise 9.30 Page 81 Math Textbook Grade 7
Problem: Given two non-perpendicular lines b, c intersecting at point A, and a point H not on b or c (H.9.47). Find points B on b and C on c such that triangle ABC has H as its orthocenter.
Solution: The orthocenter is the intersection of three altitudes in a triangle.
Answer:
Draw line d passing through H and perpendicular to line c. Let B be the intersection point of lines d and b.
Draw line d' through H and perpendicular to b. Let C be the intersection point of lines d' and c. Connect B to C. Then, H is the intersection of the altitudes from vertex B and vertex C of triangle ABC, so H is the orthocenter of triangle ABC.
Here is the solution to Grade 7 Math page 81 volume 2. Students can also refer to Grade 7 Math page 83 volume 2 and review Grade 7 Math page 76 volume 2 to reinforce their knowledge.
- Grade 7 Math page 83 volume 2 in the Connect Knowledge series - Concentrated practice page 83
- Grade 7 Math page 76 volume 2 in the Connect Knowledge series - Exercise 34: The concurrency of three medians, three angle bisectors in a triangle
