Question

Draw a triangle of circum radius 3 centimetres and two of the angles 75° and 50°.

Hint:

To construct a triangle with a given circumradius and angles, the key is to convert the triangle's angles into the central angles subtended by the sides. Always remember: Central Angle = 2 × Inscribed Angle.

Answer 1

We must construct a triangle given its circumradius (the radius of the circle that passes through all three vertices) and two of its angles.

The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circle. We will use this principle to construct the central angles corresponding to the triangle's sides.

1. Find the third angle of the triangle:
The sum of angles in a triangle is 180°.
Third angle = 180^ - (75^ + 50^) = 180^ - 125^ = 55^.
The triangle's angles are 50°, 55°, and 75°.
2. Find the corresponding central angles:
- Central angle opposite the 50° vertex = 2 × 50^ = 100^. - Central angle opposite the 55° vertex = 2 × 55^ = 110^. - Central angle opposite the 75° vertex = 2 × 75^ = 150^. (Check: 100^ + 110^ + 150^ = 360^).

1. Draw the Circumcircle:
Using a compass, draw a circle with a radius of 3 cm. Mark the center as O.
2. Construct the Central Angles:
a. Draw any radius OA.
b. Using a protractor centered at O, measure 100° from OA and draw a second radius OB.
c. From OB, measure 110° and draw a third radius OC. The remaining angle COA will be 150°.
3. Draw the Triangle:
Join the points A, B, and C on the circumference.
4. Result:
ABC is the required triangle. If measured, C (opposite arc AB) will be 50°, A (opposite arc BC) will be 55°, and B (opposite arc AC) will be 75°.