construct a triangle ABC similar to given triangle equilateral triangle PQR with side sum such that each of its sides is 6/7 of the corresponding angle PQR
Review how to divide a line segment into 7 equal pieces.
Do that for each side, then take 6 of them for the new sides.
To construct a triangle ABC that is similar to an equilateral triangle PQR with a side sum such that each of its sides is 6/7 of the corresponding side in triangle PQR, follow these steps:
1. Draw a line segment PQ, which represents one side of the equilateral triangle PQR.
2. Bisect the line segment PQ at point M. To do this, use a compass to draw arcs from points P and Q that intersect above and below the line segment PQ. Label the points of intersection as X and Y, respectively. Then draw a straight line connecting points X and Y. The midpoint of this line is point M.
3. Draw a line segment MX starting from point M. This will be the height of the equilateral triangle PQR. Use a compass to mark off six equal parts along the line segment MX.
4. From the sixth part of the line segment MX, draw a perpendicular line segment of length 6/7 times the corresponding side in triangle PQR. This will be the line segment AB. The 6/7 proportion can be achieved by using a compass to measure the desired length on the straight edge of the compass from the sixth part of the line segment MX, and then transferring it to point B on the line segment MX.
5. From point B, draw a line segment BC equal in length to side AB. To do this, use a compass to measure the length of side AB and transfer it to point C on the line segment MX.
6. Connect points A and C to complete triangle ABC.
Now, triangle ABC will be similar to equilateral triangle PQR with side lengths in a 6/7 proportion.