A parallelogram equal to a triangle
Construct △ABC with BC = 6.5 cm, ∠ABC = 120° and AB = 4 cm. Then construct a parallelogram PBDQ of the same area with PB = 5 cm.
Construct △ABC with BC = 7 cm, AB = 6 cm and AC = 5 cm. Then construct a parallelogram PBDQ of the same area with ∠PBD = 60°.
Construct △ABC with BC = 6 cm, ∠ABC = 60° and AB = 5 cm. Then construct a parallelogram PBDQ of the same area (a rectangle).
Construct △ABC with BC = 6.4 cm, AB = 4.5 cm and AC = 5.5 cm. Then construct a parallelogram PBDQ of the same area with PB = 5.5 cm.
Construct △ABC with BC = 6 cm, ∠ABC = 75° and ∠ACB = 45°. Then construct a parallelogram PBDQ of the same area with PB = 6 cm.
1. Draw BC = 6.5 cm.
2. At B, construct ∠ABC = 120° and cut BA = 4 cm on its arm. Join AC.
3. Through A, draw ST ∥ BC by copying ∠ABC at A, so that ∠BAS = ∠ABC.
4. Bisect BC: the perpendicular bisector cuts it at its midpoint D.
5. With B as the center and radius 5 cm, cut ST at P.
6. With P as the center and radius BD, cut PT at Q. Join BP and DQ: PBDQ is the parallelogram.
1. Draw BC = 7 cm.
2. With B as the center and radius 6 cm, and C as the center and radius 5 cm, draw arcs that cut at A. Join AB and AC.
3. Through A, draw ST ∥ BC by copying ∠ABC at A, so that ∠BAS = ∠ABC.
4. Bisect BC: the perpendicular bisector cuts it at its midpoint D.
5. At B, construct ∠DBP = 60°; its arm meets ST at P.
6. With P as the center and radius BD, cut PT at Q. Join BP and DQ: PBDQ is the parallelogram.
1. Draw BC = 6 cm.
2. At B, construct ∠ABC = 60° and cut BA = 5 cm on its arm. Join AC.
3. Through A, draw ST ∥ BC by copying ∠ABC at A, so that ∠BAS = ∠ABC.
4. Bisect BC: the perpendicular bisector cuts it at its midpoint D.
5. At B, construct ∠DBP = 90°; its arm meets ST at P.
6. With P as the center and radius BD, cut PT at Q. Join BP and DQ: PBDQ is the parallelogram.
1. Draw BC = 6.4 cm.
2. With B as the center and radius 4.5 cm, and C as the center and radius 5.5 cm, draw arcs that cut at A. Join AB and AC.
3. Through A, draw ST ∥ BC by copying ∠ABC at A, so that ∠BAS = ∠ABC.
4. Bisect BC: the perpendicular bisector cuts it at its midpoint D.
5. With B as the center and radius 5.5 cm, cut ST at P.
6. With P as the center and radius BD, cut PT at Q. Join BP and DQ: PBDQ is the parallelogram.
1. Draw BC = 6 cm.
2. At B, construct an angle of 75°, and at C an angle of 45°. Their arms meet at A.
3. Through A, draw ST ∥ BC by copying ∠ABC at A, so that ∠BAS = ∠ABC.
4. Bisect BC: the perpendicular bisector cuts it at its midpoint D.
5. With B as the center and radius 6 cm, cut ST at P.
6. With P as the center and radius BD, cut PT at Q. Join BP and DQ: PBDQ is the parallelogram.
Why it works
AD is a median, so △ABD is half of △ABC. △ABD is also half of the parallelogram PBDQ: the same base BD, between BD and ST. So PBDQ = 2 × △ABD = △ABC.