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Zorluk: OrtaTriangles: Properties, Perimeter, and Area

In triangle PQRPQR, point SS lies on segment QRQR such that segment PSPS is perpendicular to QRQR. The length of altitude PSPS is 88 units. If the area of triangle PQRPQR is 5656 square units and the ratio of the area of triangle PQSPQS to the area of triangle PSRPSR is 3:43 : 4, what is the length of side PQPQ?

  1. 1010Cevap
  2. B
    4134\sqrt{13}
  3. C
    174.25\sqrt{174.25}
  4. D
    454\sqrt{5}
  5. E
    1414

Cevap

10
The area of triangle PQRPQR is given as 56 square units and altitude PS=8PS = 8. Using Area=12×QR×8\text{Area} = \frac{1}{2} \times QR \times 8, we find QR=14QR = 14. Because triangles PQSPQS and PSRPSR share altitude PSPS, the ratio of their areas equals the ratio of their bases QS:SR=3:4QS : SR = 3 : 4. Dividing QR=14QR = 14 into 7 equal parts yields QS=6QS = 6. In right-angled triangle PQSPQS, legs are PS=8PS = 8 and QS=6QS = 6, giving hypotenuse PQ=82+62=10PQ = \sqrt{8^2 + 6^2} = 10.

Adım Adım Çözüm

1
Calculate the total length of base QRQR using the area of triangle PQRPQR.
Area=12×base×height  ⟹  56=12×QR×8  ⟹  56=4×QR  ⟹  QR=14\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \implies 56 = \frac{1}{2} \times QR \times 8 \implies 56 = 4 \times QR \implies QR = 14.
The area formula for any triangle relates base, height, and total area.
2
Determine the length of segment QSQS using the given area ratio.
Since triangles PQSPQS and PSRPSR share the same altitude PSPS, their areas are proportional to their base lengths QSQS and SRSR. Therefore, QS:SR=3:4QS : SR = 3 : 4. The total parts are 3+4=73 + 4 = 7. Thus, QS=14×37=6QS = 14 \times \frac{3}{7} = 6.
Triangles sharing a common altitude have areas proportional to their respective bases.
3
Apply the Pythagorean theorem in right triangle PQSPQS to find hypotenuse PQPQ.
PQ2=PS2+QS2=82+62=64+36=100  ⟹  PQ=10PQ^2 = PS^2 + QS^2 = 8^2 + 6^2 = 64 + 36 = 100 \implies PQ = 10.
Segment PSPS is perpendicular to QRQR, forming right-angled triangle PQSPQS with legs of length 8 and 6.

Anahtar Kavram

Area of triangles, common altitude area ratio, and the Pythagorean theorem.
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Konular

Triangles: Properties, Perimeter, and Area

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