RBSE Solutions Class 9 Maths Chapter 8 Construction of Triangles Exercise 8.1

Case I: To Construct a Triangle When All the Three Sides Are Given.

Question 1. Construct a triangle ABC where AB = 4 cm, BC = 5 cm and CA = 6 cm.
Answer: Here are the steps to construct the triangle:
1. First, draw a line segment BC that is 5 cm long.
2. Next, place the compass at point B and draw an arc with a radius of 4 cm.
3. Then, place the compass at point C and draw another arc with a radius of 6 cm. This arc should cross the first arc at a point, which we will call A. This method ensures all sides are the correct length.
4. Finally, join point A to B and point A to C.
5. The triangle ABC is the required triangle.
In simple words: First draw the longest side. Then, from each end of this side, draw arcs with the lengths of the other two sides. The point where these arcs meet forms the third corner of your triangle.

🎯 Exam Tip: Always start by drawing the longest side as the base, as this often makes it easier to position the arcs for the other two sides accurately.

Question 2. Two points A and B are at a distance of 6.5 cm. Find a point C at a distance of 7 cm from A and 6 cm from B.
Answer: Here are the steps to find point C and construct the triangles:
1. Draw a line segment AB with a length of 6.5 cm.
2. Now, from point A, use your compass to draw an arc with a radius of 7 cm.
3. Next, from point B, use your compass to draw another arc with a radius of 6 cm. These two arcs will intersect at two points, one above the line AB and one below it. These points are C and C'. This shows that two such triangles can be formed, mirror images of each other.
4. Join A to C, B to C, A to C', and B to C'.
5. Therefore, \(\Delta ABC\) and \(\Delta ABC'\) are the required triangles.
In simple words: Draw a line for AB. From A, draw a big arc. From B, draw another big arc. Where they cross above and below the line are your two possible points for C.

🎯 Exam Tip: Remember that two distinct points of intersection occur unless the arcs are tangent or do not intersect, leading to two possible triangles.

Question 3. Construct a \(\Delta ABC\) where a = 6.5 cm, b = 7.2 cm, c = 8 cm. Draw the bisector of \(\angle B\) which meets AC at the point M.
Answer: To construct \(\Delta ABC\) and its angle bisector:
1. First, draw the side AC, which is 7.2 cm long. (Remember, 'b' is the side opposite angle B).
2. Next, with A as the center, use a compass to draw an arc with a radius of 8 cm. (This is side 'c', opposite angle C).
3. Then, with C as the center, draw another arc with a radius of 6.5 cm. (This is side 'a', opposite angle A). This arc will intersect the first arc, and their intersection point is B.
4. Join points A to B and C to B. This forms the required \(\Delta ABC\). An angle bisector divides the angle into two equal parts.
5. Now, draw a line that bisects \(\angle B\). Extend this line until it meets the side AC at a point, which we will call M.
In simple words: Draw one side first. Then use compasses from each end to find the third point. After making the triangle, split angle B exactly in half with a line that goes to the opposite side.

🎯 Exam Tip: Clearly label all vertices and the angle bisector to ensure full marks. The bisector should appear to divide the angle equally.

Question 4. Construct a triangle ABC where a = 1 cm, b = 5 cm and c = 4 cm. Draw perpendicular from A on BC.
Answer: To construct \(\Delta ABC\) and the perpendicular from A to BC:
1. First, draw the line segment BC, which is 1 cm long. (Side 'a' is opposite vertex A).
2. Next, with B as the center, draw an arc with a radius of 4 cm. (Side 'c' is opposite vertex C).
3. Then, with C as the center, draw another arc with a radius of 5 cm. (Side 'b' is opposite vertex B). This arc will intersect the first arc at a point, which is A.
4. Join A to B and A to C. This completes the triangle \(\Delta ABC\). The construction of the triangle is the foundation for the next step.
5. Finally, draw a line from point A that is perpendicular to BC. Let this perpendicular line meet BC at point P.
6. Join AB, AC, and AP.
7. Thus, \(\Delta ABC\) is the required triangle, and AP is the perpendicular from A to BC.
In simple words: Draw the smallest side first. From its ends, draw arcs with the other two side lengths to find the third corner. Then, draw a straight line from that third corner down to the base so it forms a square corner.

🎯 Exam Tip: Ensure your arcs are clear and intersect precisely to accurately locate vertex A. The perpendicular line must form a 90-degree angle with the base.

Question 5. Construct an equilateral triangle whose sides are of length 5.5 cm.
Answer: Here are the steps to construct an equilateral triangle:
1. Start by drawing a line segment BC that is 5.5 cm long.
2. Next, place the compass at point B and draw an arc with a radius of 5.5 cm.
3. Then, place the compass at point C and draw another arc with the same radius of 5.5 cm. These arcs will intersect at a point, which you will label A. All sides of an equilateral triangle are equal, simplifying the construction.
4. Finally, join point A to B and point A to C.
5. The triangle ABC is the required equilateral triangle.
In simple words: Draw one side. Use a compass with that same side length, draw an arc from each end. Where the arcs meet is the third corner. Connect the dots.

🎯 Exam Tip: In an equilateral triangle, all sides are equal and all angles are 60 degrees. This construction relies on that fundamental property.

Question 6. Construct an isosceles triangle whose base is of length 3 cm and the other sides are of 5 cm each.
Answer: Here are the steps to construct the isosceles triangle:
1. First, draw the base line segment BC, which is 3 cm long.
2. Next, place your compass at point B and draw an arc with a radius of 5 cm.
3. Then, place your compass at point C and draw another arc with the same radius of 5 cm. These two arcs will intersect at a point, which you should label A. An isosceles triangle has two equal sides, which are used here.
4. Finally, join point B to A and point C to A.
5. The triangle ABC is the required isosceles triangle.
In simple words: Draw the base. From each end of the base, draw an arc using the length of the equal sides. The point where these arcs cross will be the top point of your triangle.

🎯 Exam Tip: Always make sure your compass is set precisely to the correct radius for accurate arc intersection, especially when drawing the equal sides.