CYNTHIA LUU AND SANDIE REYES Geometry TRIANGLES & CIRCLES CYNTHIA LUU AND SANDIE REYES THURSDAY JUNE 13TH 2013 902

Outcomes Triangles: E3-Make informal deductions using congruent triangle and angle properties E4-Demonstrate an understanding of and apply the properties of similar triangles Circles: E6-Recognize, name, describe, and represent arcs, chords, tangents, central angles, inscribed angles and circumscribed angles and make generalizations about their relationships in circles

Unique Triangles To tell if a triangle is unique (congruent) or similar you need to know certain sides and angles. To tell if it’s a unique triangle you need either: All three sides (SSS) Two sides and an angle that is enclosed by the two sides. (SAS) Two sides and a right angle (the RHS rule: Right angled, Hypotenuse, Side). Two angles and a side. (AAS) It’s not a unique triangle if you only know: Three angles (AAA) Two sides and an angle (SSA, ASS)

Two triangles are congruent if they have: exactly the same three sides and exactly the same three angles. SSS ( side, side, side) SAS (side, angle, side) ASA (angle, side, angle) RHS ( Right angled, Hypotenuse, Side). Is congruent to: Is congruent to: Is congruent to: Is congruent to:

Similar Triangles Unique (Congruent) = Similar To tell if it’s a similar triangle you need to know either: All three sides (SSS) Two sides and an angle that is enclosed by the two sides. (SAS) Two angles and a side. (AAS) Two angles or all three angles (AAA) Because all three angles add up to 180 you only need to know two angles to get the third. It’s not a similar triangle if you only know: Two sides and an angle (SSA, ASS) Unique (Congruent) = Similar Similar Unique (Congruent)

Proving Similarity Two triangles are similar if they have: all their angles equal corresponding sides are proportional SSS (three sides are in the same proportion.) SAS (Two sides are in the same proportion, and their included angle is equal.)

How to tell if Triangles are Similar To determine if the triangles shown are similar, compare their corresponding sides Are the corresponding sides proportional? Take the cross product to get the equation. Solve the equation

Questions!!!

6/10 12/x Find the value of x. 6(x)=10(12) = x 20=x

Find the value of x and y. 8/10 6/x 9/y 6/8 8(x)=10(6) 8(9)=6(y) = x

Find the value of x. 4/5 x/3 4(3)=5(x) =x 2.4=x

How tall is the tree? 2/3 x/30 2(30)=3(x) =x 20=x

x/16 5/6 16(5)=x(6) =13.33

Andy planted two trees, he planted one tree (tree A) earlier than the other. He found out the height and the length of the shadow tree B casts. He wants to know the height of the tree A. If he knows the length of the shadow tree A cast, How tall is tree A? x/17.5 10.5/11.25 17.5(10.5)=x(11.25) =x 16.33=x Tree A Tree B

Circles: Special Lines and Segments Major and minor arcs are located on the circumference of the circle. Minor Arc: A minor arc is an arc that is less than a semicircle. Major Arc B Minor Arc B A A C C Major Arc: A major arc is an arc that is greater than a semicircle.

Circles: Special Lines and Segments Tangent: A line that touches the outside of the circle at just one point The point of tangency. Chord: A chord is a line segment that joins two points within a circle. C B A B A Radius NA is perpendicular to tangent AB. The radius of the circle is always perpendicular to the tangent line. The diameter of a circle is a type of chord.

Circles: Angles in Circles Inscribed Angle: An inscribed angle is an angle whose vertex is on the circumference of the circle. 110˚ 32˚ Central Angle: A central angle is when two radii are joined the center of the circle to form an angle.

Inscribed and Central Angle The central angle is twice the inscribed angle, or the inscribed angle is half the central angle. Find the angle measurement of the central angle. If angle A measures 36˚. What does angle B measure? Angle B measures 72˚ 36 + 36 = 72

Inscribed and Central Angle If angle DAB measures 98˚. What does angle DCB measure? Angle DCB measures 49˚ 98 / 2 = 49

Circles: Angles in Circles Circumscribed Angle: A circumscribed angle is an angle with both arms tangent to the circle. A B O The relation between the central angle and the circumscribed angle is the same as to a tangent line and the radius. The radius OA is perpendicular to the tangent AB. The same thing goes for radius OC to the tangent CB. C

Sources http://www.mathsisfun.com/ http://www.bbc.co.uk Textbook (Mathematics 9) Textbook (Geometry Supplement) http://library.thinkquest.org http://www.homeschoolmath.net