Examkin

Soru

Zorluk: ZorCircle Geometry

In a circle, chords ABAB and CDCD intersect perpendicularly at point EE. If AE=35AE = 35, EB=5EB = 5, and CE=5CE = 5, what is the length of the diameter of the circle?

Cevap: 50

Cevap

50
The correct answer is 50. By applying the intersecting chords theorem, the segment EDED is found to be 35. Since the chords are perpendicular and intersect at point EE, we can find the distance from the center of the circle to each chord by analyzing the distances from the intersection point to the midpoints of the chords. The midpoints of both chords are 20 units from their endpoints. The distance from EE to the midpoint of ABAB is 35−20=1535 - 20 = 15. This distance is equal to the perpendicular distance from the center of the circle to the other chord, CDCD. Using the Pythagorean theorem with a chord half-length of 20 and a distance from the center of 15, the radius of the circle is 152+202=25\sqrt{15^2 + 20^2} = 25. Therefore, the diameter of the circle is 2×25=502 \times 25 = 50.

Adım Adım Çözüm

1
Find the length of segment EDED using the intersecting chords theorem.
ED=35ED = 35
For any two intersecting chords ABAB and CDCD intersecting at point EE, the product of the segments of one chord equals the product of the segments of the other: AE⋅EB=CE⋅EDAE \cdot EB = CE \cdot ED. Substituting the given values: 35⋅5=5⋅ED35 \cdot 5 = 5 \cdot ED, which simplifies to ED=35ED = 35.
2
Calculate the total lengths of chords ABAB and CDCD and determine their midpoints.
Chord lengths AB=40AB = 40 and CD=40CD = 40. Midpoint distances MB=20MB = 20 and ND=20ND = 20.
The total length of chord ABAB is AE+EB=35+5=40AE + EB = 35 + 5 = 40. The perpendicular line from the center OO to ABAB bisects the chord at midpoint MM, so MB=40/2=20MB = 40 / 2 = 20. Similarly, the total length of chord CDCD is CE+ED=5+35=40CE + ED = 5 + 35 = 40, and its midpoint NN bisects it, so ND=20ND = 20.
3
Find the perpendicular distances from the center OO to the chords ABAB and CDCD.
OM=15OM = 15 and ON=15ON = 15
The distance from the intersection point EE to the midpoint MM along chord ABAB is AE−AM=35−20=15AE - AM = 35 - 20 = 15. Because the chords are perpendicular, the perpendicular distance from the center OO to chord ABAB is equal to the distance ENEN along the other chord, so OM=EN=15OM = EN = 15. Similarly, ON=EM=15ON = EM = 15.
4
Calculate the radius of the circle using the Pythagorean theorem.
Radius R=25R = 25
In the right triangle OMBOMB, the hypotenuse is the radius R=OBR = OB, and the legs are the perpendicular distance OM=15OM = 15 and half the chord length MB=20MB = 20. By the Pythagorean theorem, R2=OM2+MB2=152+202=225+400=625R^2 = OM^2 + MB^2 = 15^2 + 20^2 = 225 + 400 = 625. Taking the square root gives R=25R = 25.
5
Calculate the diameter of the circle.
Diameter = 5050
The diameter of a circle is twice its radius: 2R=2⋅25=502R = 2 \cdot 25 = 50.

Anahtar Kavram

Using perpendicular chords, the intersecting chords theorem, and the Pythagorean theorem to determine the radius and diameter of a circle.
Bu soruyu puanla

Konular

Circle Geometry

Bu konudan daha fazla soru