Plane and solid geometry . from center prove chord AB > chord CD,The proof is left as an exercise for the student. Hint. Begin with A OGF. 311. Cor. I. A diameter is greater than any otherchord, 312. Cor. II. T1w locus of tlxe mid-points of all chordsof a circle equal to a given chord is tl%e circumference hav-ing the same center as tlie given circle, and having for ra-dius the perpendicular from the center to the given chord. Ex. 433. Prove Prop. IX by the indirect method. Ex. 434. Through a given point within a circle construct the mini-mum chord. Ex. 435. If two chords are drawn fro
Plane and solid geometry . from center prove chord AB > chord CD,The proof is left as an exercise for the student. Hint. Begin with A OGF. 311. Cor. I. A diameter is greater than any otherchord, 312. Cor. II. T1w locus of tlxe mid-points of all chordsof a circle equal to a given chord is tl%e circumference hav-ing the same center as tlie given circle, and having for ra-dius the perpendicular from the center to the given chord. Ex. 433. Prove Prop. IX by the indirect method. Ex. 434. Through a given point within a circle construct the mini-mum chord. Ex. 435. If two chords are drawn from one extremity of a diameter,making unequal angles with it, the chords are unequal. Ex. 436. The perpendicular from the center of a circle to a side ofan inscribed equilateral triangle is less than the perpendicular from thecenter of the circle to a side of an inscribed square. (See § 126 PLANE GEOMETRi Proposition X. Theorem 313. A tangent to a circle is perpendicular to tlie ravinsdrawn to tlw point of A M T B Given line AB, tangent to circle 0 at r, and OT, a radius drawnto the point of prove AB ± OT. Argument 1. Let M be any point on AB other than T; then M is outside the circumference. 2. Draw OM, intersecting the circumference at S, 3. OS < OM, 4. OS = OT, 5. .-. OT < or is the shortest line that can be drawn from 0 to 0T± AB ; AB ± OT 6. 7. Reasons 1. § 286. 2. § 54, 15. 3. § 54, 12. 4. § 279, a. 5. § 309. 6. Arg. 5. i. § 165. 314. Cor. I. (Converse of Proj). X). A straight line per])endicular to a radius at its outer extremity is tangent to the circle. Hint. Prove by the indirect method. In tlie figure for Prop. X,suppose that AB i^ not tangent to circle 0 at point T; then draw CDthrough T, tangent to circle 0. Apply §03. 315. Cor. II. A perpendicular to a tangent at the pointof tangcncij passes t]trough the center oftlie circle.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
