Samacheer Kalvi Class 9 Maths Solutions Chapter 4 Geometry Exercise 4.6

 

Question 1. Draw a triangle ABC, where AB = 8 cm, BC = 6 cm and \( \angle B = 70^\circ \) and locate its circumcentre and draw the circumcircle.
Answer:

The steps to construct the circumcircle are as follows:
1. First, draw triangle ABC using the given measurements. Draw segment AB 8 cm long. At point B, draw a line at a 70-degree angle. On this line, mark point C such that BC is 6 cm. Join points A and C to complete the triangle.
2. Next, construct the perpendicular bisectors for any two sides of the triangle, such as AB and BC. These bisectors are lines that cut each side exactly in half and are at a 90-degree angle to them. The point where these two bisectors meet is called S, which is the circumcenter.
3. Now, with S as the center, draw a circle using SA, SB, or SC as the radius. This circle, called the circumcircle, will pass through all three vertices of the triangle: A, B, and C. A circumcircle is a circle that passes through all the vertices of a polygon.
The measured circum radius is 4.3 cm.
In simple words: First, draw the triangle with the given side lengths and angle. Then, find the center point (circumcenter) by drawing lines that cut two sides in half at a right angle. Finally, draw a circle from this center point that touches all three corners of the triangle.

🎯 Exam Tip: Always use a sharp pencil and precise measurements with a compass and ruler to ensure your construction is accurate for full marks.

 

Question 2. Construct the right triangle PQR whose perpendicular sides are 4.5 cm and 6 cm. Also locate its circumcentre and draw circumcircle.
Answer:

The steps to construct the circumcircle are as follows:
1. First, draw the right-angled triangle PQR using the given measurements. Draw segment PQ 6 cm long. At point Q, draw a line perpendicular to PQ. On this line, mark point R such that QR is 4.5 cm. Connect points P and R to complete the triangle.
2. Next, construct the perpendicular bisectors for any two sides of the triangle, such as PQ and PR. These bisectors are lines that cut each side exactly in half and are at a 90-degree angle to them. The point where these two bisectors meet is called S, which is the circumcenter. For a right-angled triangle, the circumcenter always lies at the midpoint of the hypotenuse.
3. Now, with S as the center, draw a circle using SP, SQ, or SR as the radius. This circle, called the circumcircle, will pass through all three vertices of the triangle: P, Q, and R.
The measured circum radius is 3.8 cm.
In simple words: First, draw the right-angled triangle. Then, find its center point by drawing lines that cut two sides in half at a right angle. Finally, draw a circle from this center point that touches all three corners of the triangle.

🎯 Exam Tip: Remember that the circumcenter of a right-angled triangle is always located at the midpoint of its hypotenuse.

 

Question 3. Construct \( \triangle ABC \) with AB = 5 cm \( \angle B = 100^\circ \) and BC = 6 cm. Also locate its circumcentre draw circumcircle.
Answer:

The steps to construct the circumcircle are as follows:
1. First, draw triangle ABC using the given measurements. Draw segment AB 5 cm long. At point B, draw a line at a 100-degree angle. On this line, mark point C such that BC is 6 cm. Join points A and C to complete the triangle.
2. Next, construct the perpendicular bisectors for any two sides of the triangle, such as AB and BC. The point where these two bisectors meet is called S, which is the circumcenter. For an obtuse triangle, the circumcenter is always outside the triangle.
3. Now, with S as the center, draw a circle using SA, SB, or SC as the radius. This circle, called the circumcircle, will pass through all three vertices of the triangle: A, B, and C.
The measured circum radius is 4.3 cm.
In simple words: Draw the triangle as instructed. Find the center (circumcenter) by making lines that cut two sides in half at a right angle. Then, draw a circle from this center that passes through all three corners of the triangle.

🎯 Exam Tip: Pay close attention to the angle measurement and ensure your protractor is aligned correctly to get an accurate 100° angle, as this affects the triangle's shape and circumcenter location.

 

Question 4. Construct an isosceles triangle PQR where PQ = PR and \( \angle Q = 50^\circ \), QR = 7cm. Also draw its circumcircle.
Answer:

The steps to construct the circumcircle are as follows:
1. First, draw the isosceles triangle PQR. Since PQ = PR, the angles opposite these sides, \( \angle Q \) and \( \angle R \), will be equal. So, if \( \angle Q = 50^\circ \), then \( \angle R = 50^\circ \). Draw segment QR 7 cm long. At points Q and R, draw lines at 50-degree angles. The point where these lines meet is P. Join PQ and PR to complete the triangle.
2. Next, construct the perpendicular bisectors for any two sides of the triangle, such as QR and PR. The point where these bisectors meet is called S, which is the circumcenter.
3. Now, with S as the center, draw a circle using SP, SQ, or SR as the radius. This circle, called the circumcircle, will pass through all three vertices of the triangle: P, Q, and R. All vertices of an isosceles triangle lie on its circumcircle.
The measured circum radius is 3.5 cm.
In simple words: Draw the isosceles triangle, making sure the base angles are equal. Then, find the center point (circumcenter) by bisecting two sides at a right angle. Finally, draw a circle from this center that touches all three corners of the triangle.

🎯 Exam Tip: For isosceles triangles, remember that the angles opposite the equal sides are also equal, which helps in constructing the triangle accurately.

 

Question 5. Draw an equilateral triangle of side 6.5 cm and locate its incentre. Also draw the incircle.
Answer:

The steps to construct the incircle are as follows:
1. First, draw the equilateral triangle ABC with each side measuring 6.5 cm. All three sides and angles will be equal.
2. Next, construct the angle bisectors of any two angles, such as \( \angle A \) and \( \angle B \). An angle bisector is a line that divides an angle into two equal parts. The point where these two angle bisectors meet is called I, which is the incenter of \( \triangle ABC \). For an equilateral triangle, the incenter, circumcenter, centroid, and orthocenter all coincide at the same point.
3. Now, draw a line from I that is perpendicular to any side of the triangle, for example, side AB. Let this perpendicular line meet AB at point D.
4. With I as the center and ID as the radius, draw a circle. This circle is the incircle, and it will touch all three sides of the triangle internally.
The measured in-radius is 1.5 cm.
In simple words: Draw the equilateral triangle. Find the center (incenter) by drawing lines that cut two angles in half. From this center, drop a straight line down to one side. This length is the radius to draw a circle that touches all three inner sides of the triangle.

🎯 Exam Tip: Remember that for an equilateral triangle, all angle bisectors are also medians, altitudes, and perpendicular bisectors, and they all meet at a single point which serves as both the incenter and circumcenter.

 

Question 6. Draw a right triangle whose hypotenuse is 10 cm and one of the legs is 8 cm. Locate its incentre and also draw the incircle
Answer:

The steps to construct the incircle are as follows:
1. First, draw the right-angled triangle ABC. Draw segment AB 8 cm long. At point A, draw a perpendicular line. From point B, draw an arc with a radius of 10 cm (the hypotenuse length) to cut the perpendicular line at point C. Join B and C. (Using Pythagoras theorem, the other leg AC would be \( \sqrt{10^2 - 8^2} = \sqrt{100 - 64} = \sqrt{36} = 6 \text{ cm} \)).
2. Next, construct the angle bisectors of any two angles, such as \( \angle B \) and \( \angle C \). The point where these two angle bisectors meet is called I, which is the incenter of \( \triangle ABC \). The incenter is always inside the triangle.
3. Now, draw a line from I that is perpendicular to any side of the triangle, for example, side AB. Let this perpendicular line meet AB at point D.
4. With I as the center and ID as the radius, draw a circle. This circle is the incircle, and it will touch all three sides of the triangle internally.
The measured in-radius is 1.8 cm.
In simple words: Draw the right-angled triangle using the given hypotenuse and one side. Then, find the center (incenter) by drawing lines that cut two angles in half. From this center, draw a perpendicular line to one side to find the radius. Finally, draw a circle from the center using this radius, which will touch all three inside edges of the triangle.

🎯 Exam Tip: For a right-angled triangle, if you know the hypotenuse and one leg, you can use the Pythagorean theorem to find the length of the other leg to cross-check your construction.

 

Question 7. Draw \( \triangle ABC \) given AB = 9 cm, \( \angle CAB = 115^\circ \) and \( \angle ABC = 40^\circ \). Locate its incentre and also draw the incircle. (Note: You can check from the above examples that the incentre of any triangle is always in its interior).
Answer:

The steps to construct the incircle are as follows:
1. First, draw triangle ABC using the given measurements. Draw segment AB 9 cm long. At point A, draw a line at a 115-degree angle (for \( \angle CAB \)). At point B, draw a line at a 40-degree angle (for \( \angle ABC \)). The point where these two lines meet is C. This will form an obtuse triangle.
2. Next, construct the angle bisectors of any two angles, such as \( \angle A \) and \( \angle B \). The point where these two angle bisectors meet is called I, which is the incenter of \( \triangle ABC \). As observed, the incenter of any triangle is always located inside the triangle.
3. Now, draw a line from I that is perpendicular to any side of the triangle, for example, side AB. Let this perpendicular line meet AB at point D.
4. With I as the center and ID as the radius, draw a circle. This circle is the incircle, and it will touch all three sides of the triangle internally.
The measured in-radius is 2.7 cm.
In simple words: Draw the triangle with the given side and angles. Find the incenter by splitting two angles in half. From this center, draw a straight line that touches one side at a 90-degree angle to get the radius. Then, draw a circle using this radius, which will touch the inside of all three sides of the triangle.

🎯 Exam Tip: Always double-check your angle measurements, especially for obtuse angles, as slight inaccuracies can significantly change the position of the incenter and the incircle's radius.

 

Question 8. Construct \( \triangle ABC \) in which AB = BC = 6cm and \( \angle B = 80^\circ \). Locate its incentre and draw the incircle.
Answer:

The steps to construct the incircle are as follows:
1. First, draw triangle ABC using the given measurements. Draw segment AB 6 cm long. At point B, draw a line at an 80-degree angle. On this line, mark point C such that BC is 6 cm. Join points A and C to complete the isosceles triangle. Since AB = BC, it is an isosceles triangle.
2. Next, construct the angle bisectors of any two angles, such as \( \angle A \) and \( \angle B \). The point where these two angle bisectors meet is called I, which is the incenter of \( \triangle ABC \). The incenter is always inside the triangle.
3. Now, draw a line from I that is perpendicular to any side of the triangle, for example, side AB. Let this perpendicular line meet AB at point D.
4. With I as the center and ID as the radius, draw a circle. This circle is the incircle, and it will touch all three sides of the triangle internally.
The measured in-radius is 1.9 cm.
In simple words: Draw the triangle with two equal sides and the angle between them. Find the center point (incenter) by drawing lines that cut two angles in half. From this center, draw a perpendicular line to one side to find the radius. Then, draw a circle that touches all three inner sides of the triangle.

🎯 Exam Tip: When constructing an isosceles triangle, ensure the two equal sides are drawn accurately from the vertex angle, or that the base angles are equal if using the base side as reference.