Chapter 9
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52 terms
Terms | Definitions |
|---|---|
 Theorm 8-1 | If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. |
| Corollary 1: | When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse. |
| Corollary 2: | When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg. |
| Pythagorean Theorem | In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. |
| Theorem 8-3 | If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. |
| Theorem 8-4 | If the square of the longest side of a triangle is less than the sum of the squares of the other sides, then the triangle is an acute triangle. |
| Theorem 8-5 | If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is an obtuse triangle. |
| Theorem 8-6 (45-45-90) | In a 45-45-90 triangle, the hypotenuse is √2 times as long as a leg. |
| Theorem 8-7 (30-60-90) | In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. |
| Theorem 9-1 | If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. |
| Corollary: | Tangents to a circle from a point are congruent. |
| Theorem 9-2 | If a line in the plane if a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle. |
| Theorem 9-3 | In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent. |
| Theorem 9-4 | In the same circle or in congruent circles; A) congruent arcs have congruent, B) congruent chords have congruent arcs. |
| Theorem 9-5 | A diameter that is perpendicular to a chord bisects the chord and its arc. |
| Theorem 9-6 | In the same circle or in congruent circles; A) chords equally distant from the center (centers) are congruent, B) congruent chords are equally distant from the center (centers). |
| Theorem 9-7 | The measure of an inscribed angle is equal to half the measure if its intercepted arc. |
| Corollary 1: | If two inscribed angles intercept the same arc, then the angles are congruent. |
| Corollary 2: | An angle inscribed in a semicircle is a right angle. |
| Corollary 3: | If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. |
| 9-8 | The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc. |
| 9-9 | the measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. |
| 9-10 | The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the intercepted arcs. |
| 9-11 | When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. |
| 9-12 | When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. |
| 9-13 | When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment. |
| 11-1 | The area of a rectangle equals the product of its base and height (A=bh) |
| 11-2 | The area of a parallelogram equals the product of a base and a height to that base. (A=bh) |
| 11-3 | The area of a triangle equals half the product of a base and the height to the base. (A=1/2bh) |
| 11-4 | The area of a rhombus equals half the product of its diagonals. (A=1/2d1d2) |
| 11-5 | The area of a trapezoid equals half the product of the height and the apothem and the perimeter. (A=1/2ap) |
| 11-6 | The area of a regular polygon is equal to half the product of the apothem and the perimeter. |
| 11-7 | If the scale factor of tow similar figures is a:b, then; 1) the ratio of the perimeters is a:b, 2) the ratio of the areas is a2:b2. |
| 12-1 | The lateral area of a right prism equals the perimeter of a base times the height of the prism. |
| 12-2 | The volume of a right prism equals the area of a base times the height of the prism. |
| 12-3 | The lateral area of a regular pyramid equals half the perimeter of the base times the slant height. |
| 12-4 | The volume of a pyramid equals one third the area of the base times the height of the pyramid. |
| 12-5 | The lateral area of a cylinder equals the circumference of a base times the height of the cylinder. |
| 12-6 | The volume of a cylinder equals the base time the height of the cylinder. |
| 12-7 | The lateral area of a cone equals half the circumference of the base times the slant height. |
| 12-8 | The volume of a cone equals 1/3 the area of the base times the height of the cone. |
| 12-9 | The area of a sphere equals 4pi times the radius. |
| 12-10 | The volume of a sphere equals 4/3pi times the cube of the radius. |
| 12-11 | The scale factor of two similar solids is a:b, then 1)the ratio of corresponding perimeters is a:b 2) the ratio of the base areas, of the lateral areas, and of the total areas is a2:b2 3) the ratio of the volumes is a3:b3 |
| 13-1 (Distance Formula) | d= rad:(x2-x1)2 + (y2-y1)2 |
| 13-2 | An equation of the circle with center (a,b) and radius r is (x-a)2 + (y-b)2= r2 |
| 13-3 | Two nonvertical lines are parallel if and only if their slopes are equal. |
| 13-4 | Two nonvertical lines are perpendicular if and only if the product of their slopes is -1 |
| 13-5 (Midpoint Formula) | The midpoint of the segment that joins points (x1, y1) and (x2, y2) |
| 13-6 (Standard Form) | The graph of any equation that can be written in the form Ax+By=C, with a and b not both zero, is a line. |
| 13-7 (Slope-Intercept) | A line with the equation y=mx+b has slope m and y-intercept b. |
| 13-8 (Point-Slope Form) | AN equation of the line that passes through the point (x1, y1) and has slope m is y-y1=m(x-x1). |
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