Question

Draw a circle of radius 2.5 centimetres. Draw a triangle of two angles 70° and 50° with its sides touches the circle.

Hint:

To construct a triangle with a given incircle and angles, the key is to find the central angles between the radii to the tangency points. Remember the rule: Central Angle = 180° - Vertex Angle. This is the reverse logic of the circumcircle construction.

Answer 1

We need to construct a triangle that is circumscribed about a given circle (the circle is the triangle's incircle). We are given the incircle's radius and two angles of the triangle.

The incenter (center of the incircle) is the meeting point of the angle bisectors. The radius to a point of tangency is perpendicular to the tangent side. In the quadrilateral formed by the incenter O, a vertex A, and two points of tangency, the angle at the center O is 180^ - A.

1. Find the third angle of the triangle:
Third angle = 180^ - (70^ + 50^) = 180^ - 120^ = 60^.
The triangle's angles are 50°, 60°, and 70°.
2. Find the central angles between radii to points of tangency:
- Central angle corresponding to the 50° vertex = 180^ - 50^ = 130^. - Central angle corresponding to the 60° vertex = 180^ - 60^ = 120^. - Central angle corresponding to the 70° vertex = 180^ - 70^ = 110^. (Check: 130^ + 120^ + 110^ = 360^).

1. Draw the Incircle:
Using a compass, draw a circle with a radius of 2.5 cm. Mark the center as O.
2. Construct the Central Angles:
a. Draw any radius OP.
b. Using a protractor centered at O, measure 130° from OP and draw a second radius OQ.
c. From OQ, measure 120° and draw a third radius OR. The remaining angle ROP will be 110°.
3. Construct the Tangents (Sides of the Triangle):
a. At point P on the circle, construct a line perpendicular to the radius OP.
b. At point Q, construct a line perpendicular to the radius OQ.
c. At point R, construct a line perpendicular to the radius OR.
4. Form the Triangle:
Extend these three perpendicular lines (tangents) until they intersect each other. The points of intersection form the vertices of the required triangle.
5. Result:
The resulting triangle will have its sides tangent to the circle and its angles will be 50°, 60°, and 70°.