RBSE Solutions Class 9 Maths Chapter 8 Construction of Triangles Exercise 8.4

Chapter 8 Construction of Triangles Ex 8.4

Question 1. Construct a right-angled triangle the length of whose hypotenuse is 5 cm and one of the remaining side is 3 cm long.
Answer:The steps to construct the right-angled triangle are:
1. First, draw a line segment AB that is 3 cm long.
2. Next, place the compass at point A and draw a right angle (90°) using a ruler and compass.
3. Then, from point A, draw a line upwards along the 90° angle.
4. Now, with point B as the center, open the compass to 5 cm (this is the length of the hypotenuse). Draw an arc that cuts the 90° angle line at a point we will call C.
5. Finally, connect point B to point C. This completes the triangle.
Thus, \( \triangle ABC \) is the required right-angled triangle, where \( \angle A = 90^\circ \), AB = 3 cm, and BC = 5 cm (hypotenuse). According to the Pythagorean theorem, the side AC would be 4 cm.

In simple words: Draw a 3 cm line, then make a 90-degree angle at one end. From the other end of the 3 cm line, draw a 5 cm arc that cuts the angle line. Connect the points to form the right-angled triangle.

🎯 Exam Tip: Always start with the given side length, then use the angle and hypotenuse/other side information to locate the third vertex. Clearly label all vertices and side lengths.

Question 2. Construct a triangle ABC when \( \angle A = 90^\circ \), AC = 5.4 cm and hypotenuse BC = 10 cm.
Answer:The steps to construct \( \triangle ABC \) are:
1. First, draw a line segment AC that is 5.4 cm long.
2. At point A, construct a ray AX perpendicular to AC, such that \( \angle CAX = 90^\circ \). This will be one side of the right angle.
3. Now, with point C as the center and a radius of 10 cm (the length of the hypotenuse BC), draw an arc that intersects the ray AX at point B.
4. Finally, connect point B to point C. This completes the triangle.
Thus, \( \triangle ABC \) is the required right-angled triangle with \( \angle A = 90^\circ \), AC = 5.4 cm, and BC = 10 cm.

In simple words: Draw a line for AC (5.4 cm). At point A, draw a straight line going up to make a 90-degree angle. From point C, draw an arc with a 10 cm radius. Where this arc cuts the line from A, call that point B. Join B and C to finish the triangle.

🎯 Exam Tip: When constructing a right-angled triangle with hypotenuse given, use the arc method from the non-right-angle vertex to find the third vertex on the perpendicular line.

Question 3. Construct a right angled triangle ABC whose \( \angle A = 90^\circ \) and a = 10 cm and c = 6 cm, from the vertex A, draw a perpendicular on the hypotenuse.
Answer:Here, side 'a' refers to the side opposite vertex A (BC), and side 'c' refers to the side opposite vertex C (AB). So, BC = 10 cm and AB = 6 cm.
The steps to construct \( \triangle ABC \) and draw the perpendicular are:
1. First, draw a line segment AB that is 6 cm long as the base.
2. With point A as the center, construct a ray AX such that \( \angle BAX = 90^\circ \). This forms the right angle at A.
3. Now, with point B as the center and a radius of 10 cm (the hypotenuse BC), draw an arc that intersects the ray AX at point C.
4. Connect point B to point C. This completes the right-angled triangle \( \triangle ABC \).
5. From vertex A, draw a line segment AD such that it is perpendicular to the hypotenuse BC, where D lies on BC. This perpendicular shows the shortest distance from A to the hypotenuse.
Thus, \( \triangle ABC \) is the required right-angled triangle, and AD is the perpendicular from A to the hypotenuse BC.

In simple words: Draw a 6 cm line for AB. At point A, draw a line straight up to make a 90-degree angle. From point B, draw an arc with a 10 cm radius that cuts the line from A; this point is C. Join B and C. Then, draw a dashed line from A straight down to the line BC so that it meets BC at a 90-degree angle; label this point D.

🎯 Exam Tip: When drawing a perpendicular from a vertex to the opposite side, ensure the line forms a 90-degree angle with that side. Use dashed lines for construction details like perpendiculars or altitudes.