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52 terms

Chapter 9

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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).