A triangle equal to a parallelogram
Construct parallelogram ABCD with BC = 4 cm, AB = 5 cm and ∠ABC = 60°. Then construct △PBE of the same area with PB = 6 cm.
Construct parallelogram ABCD with BC = 5 cm, AB = 4 cm and ∠ABC = 120°. Then construct △PBE of the same area right-angled at B.
Construct parallelogram ABCD with BC = 4.5 cm, AB = 4 cm and ∠ABC = 45°. Then construct △PBE of the same area with PB = 5 cm.
Construct parallelogram ABCD with BC = 3.5 cm, AB = 4.5 cm and ∠ABC = 75°. Then construct △PBE of the same area with ∠PBE = 60°.
1. Draw BC = 4 cm.
2. At B, construct ∠ABC = 60° and cut BA = 5 cm on its arm.
3. With A as the center and radius 4 cm, and C as the center and radius 5 cm, draw arcs that cut at D. Join AD and CD: ABCD is the given parallelogram.
4. Produce BC to E, making CE = BC: with C as the center and radius 4 cm, cut it at E.
5. Produce AD both ways to make the line ST.
6. With B as the center and radius 6 cm, cut ST at P.
7. Join PB and PE: △PBE has the same area as ABCD.
1. Draw BC = 5 cm.
2. At B, construct ∠ABC = 120° and cut BA = 4 cm on its arm.
3. With A as the center and radius 5 cm, and C as the center and radius 4 cm, draw arcs that cut at D. Join AD and CD: ABCD is the given parallelogram.
4. Produce BC to E, making CE = BC: with C as the center and radius 5 cm, cut it at E.
5. Produce AD both ways to make the line ST.
6. At B, construct ∠PBE = 90°; its arm meets ST at P.
7. Join PB and PE: △PBE has the same area as ABCD.
1. Draw BC = 4.5 cm.
2. At B, construct ∠ABC = 45° and cut BA = 4 cm on its arm.
3. With A as the center and radius 4.5 cm, and C as the center and radius 4 cm, draw arcs that cut at D. Join AD and CD: ABCD is the given parallelogram.
4. Produce BC to E, making CE = BC: with C as the center and radius 4.5 cm, cut it at E.
5. Produce AD both ways to make the line ST.
6. With B as the center and radius 5 cm, cut ST at P.
7. Join PB and PE: △PBE has the same area as ABCD.
1. Draw BC = 3.5 cm.
2. At B, construct ∠ABC = 75° and cut BA = 4.5 cm on its arm.
3. With A as the center and radius 3.5 cm, and C as the center and radius 4.5 cm, draw arcs that cut at D. Join AD and CD: ABCD is the given parallelogram.
4. Produce BC to E, making CE = BC: with C as the center and radius 3.5 cm, cut it at E.
5. Produce AD both ways to make the line ST.
6. At B, construct ∠PBE = 60°; its arm meets ST at P.
7. Join PB and PE: △PBE has the same area as ABCD.
Why it works
BE is twice BC, and P is on ST, the line of AD, so △PBE and ABCD have the same height. A triangle has half the area of a parallelogram on the same base and of the same height, and BE is twice the base, so the two areas are equal.