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Zorluk: ZorProperties of Quadrilaterals

In rhombus ABCDABCD, the perimeter is 100100 and the length of diagonal BDBD is 3030. Point PP lies on diagonal ACAC such that the ratio of the length of segment APAP to the length of segment PCPC is 3:73:7. What is the length of segment BPBP?

Cevap: 17

Cevap

The length of segment BPBP is 17.
The correct answer is found by utilizing the properties of a rhombus. A rhombus has four congruent sides, meaning each side of a rhombus with perimeter 100100 has a length of 2525. The diagonals of a rhombus are perpendicular bisectors of one another. Letting OO be the intersection of the diagonals, we find BO=15BO = 15 since diagonal BD=30BD = 30. Using the Pythagorean theorem on right triangle AOBAOB, we determine that the other half-diagonal is AO=252−152=20AO = \sqrt{25^2 - 15^2} = 20, which means the full diagonal AC=40AC = 40. Point PP divides ACAC in the ratio 3:73:7, meaning AP=12AP = 12 and PC=28PC = 28. The distance from PP to the intersection point OO is OP=AO−AP=20−12=8OP = AO - AP = 20 - 12 = 8. Finally, applying the Pythagorean theorem to right triangle BOPBOP with legs BO=15BO = 15 and OP=8OP = 8 yields BP=152+82=17BP = \sqrt{15^2 + 8^2} = 17.

Adım Adım Çözüm

1
Calculate the side length of rhombus ABCDABCD from its perimeter.
Each side length is 2525.
A rhombus has four equal sides, so the side length is the perimeter divided by four: 1004=25\frac{100}{4} = 25.
2
Find the length of half of diagonal BDBD.
BO=15BO = 15, where OO is the intersection of diagonals ACAC and BDBD.
The diagonals of a rhombus bisect each other.
3
Calculate the half-diagonal length AOAO and full diagonal length ACAC.
AO=20AO = 20 and AC=40AC = 40.
The diagonals of a rhombus are perpendicular, forming right triangle AOBAOB. By the Pythagorean theorem, AO=AB2−BO2=252−152=20AO = \sqrt{AB^2 - BO^2} = \sqrt{25^2 - 15^2} = 20. Since the diagonals bisect each other, the total length of diagonal ACAC is 2×20=402 \times 20 = 40.
4
Determine the length of segment APAP.
AP=12AP = 12.
Point PP lies on diagonal ACAC such that the ratio of segment APAP to PCPC is 3:73:7. Therefore, AP=33+7×AC=310×40=12AP = \frac{3}{3+7} \times AC = \frac{3}{10} \times 40 = 12.
5
Find the distance OPOP between point PP and the intersection point OO.
OP=8OP = 8.
Since AO=20AO = 20 and PP is 1212 units from AA, PP lies on the segment AOAO. Thus, the distance from PP to OO is OP=AO−AP=20−12=8OP = AO - AP = 20 - 12 = 8.
6
Calculate the length of segment BPBP.
BP=17BP = 17.
Because the diagonals of a rhombus are perpendicular, △BOP\triangle BOP is a right triangle with legs BO=15BO = 15 and OP=8OP = 8. Using the Pythagorean theorem, BP=BO2+OP2=152+82=17BP = \sqrt{BO^2 + OP^2} = \sqrt{15^2 + 8^2} = 17.

Anahtar Kavram

Properties of Rhombuses (perpendicular bisecting diagonals, equal side lengths) and the Pythagorean Theorem
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Konular

Properties of Quadrilaterals

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