Geometry Exam Notes

Circumcenter and Algebra Review

Circumcenter

• The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect.

• It is the center of the circumcircle of the triangle.

• Location:

  • Acute triangle: Inside the triangle.

  • Obtuse triangle: Outside the triangle.

  • Right triangle: On the midpoint of the hypotenuse.

Finding the Circumcenter

1. Given a triangle with vertices A, B, and C:

   • A $(-3, -4)$, B $(1, 7)$, C $(8, -9)$.

2. Write the equation of the perpendicular bisector of AB:

   • Step 1: Find the midpoint of AB.

   • Step 2: Find the slope of AB.

   • Step 3: Write the equation of the perpendicular bisector of AB.

3. Find the equation of the perpendicular bisector of BC.

4. Solve the system of equations formed by the equations from steps 2 and 3 to find the coordinates of the circumcenter.

5. Find the equation of the perpendicular bisector of AC. Use this to check the answer from step 4.

Additional Problem

• Given triangle DDOG with vertices:

  • D $(13, -1)$, O $(-9, 3)$, G $(-3, -9)$.

• Follow the same steps as above to find the circumcenter.

Algebra Review

Factor Completely

1. $24x^3 – 16x^2$

2. $3x^2 + 11x – 4$

3. $4x^2 – 49$

4. $2x^2 + 12x - 54$

Simplify

• Show all work and give exact answers (no calculators).

1. $\sqrt{63}$

2. $5\sqrt{48} + 5\sqrt{32} − 3\sqrt{27}$

3. $3\sqrt{12} \cdot 5\sqrt{10}$

4. $\frac{15}{6} \sqrt{\frac{80}{5}}$

Solve

• Show all work and give exact answers (no calculators).

1. $\frac{1}{3}(x − 2) = 4 − \frac{x}{2}$

2. $(x − 3)^2 = 20$

Guided Notes: HONORS GEOMETRY Ch. 4 Perpendicular Bisectors and Circumcenter

• Perpendicular Bisector: A line segment that bisects another line segment at a right angle.

• Circumcenter: The point of intersection of the perpendicular bisectors of the sides of a triangle.

  • The circumcenter is equidistant from the vertices of the triangle.

  • The circumcenter is the center of the circumcircle of the triangle.

  • Denoted by P(X, Y).

  • Can be inside, outside, or on the triangle (acute, obtuse, or right triangles, respectively).

How to find the circumcenter:

1. Find the midpoint of each side of the triangle.

2. Find the slope of each side of the triangle.

3. Find the perpendicular slope to write the equation of the bisector (Bisector Islope).

4. Do this twice.

5. Use a calculator to find the intersection.

Review: Midpoint, Slope, and Perpendicular Slope

• Midpoint: Given points $(4, 8)$ and $(-2, 10)$, find the midpoint of segment AB.

• Slope: Find the slope of segment AB.

• Perpendicular Slope: What is the perpendicular slope of segment AB?

• Equation of Line: What is the equation of the line perpendicular to segment AB?

Example 1: Finding the Equation of the Perpendicular Bisector

• Given points J$(5, 7)$ and B $(-6, 12)$.

• Find the equation of the perpendicular bisector of JB.

• Midpoint $M = (\frac{x<em>1 + x</em>2}{2}, \frac{y<em>1 + y</em>2}{2})$

• Slope $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

Example 2: Finding the Circumcenter

• Given triangle ABC with vertices at A $(-3, −4)$, B $(1, 7)$, C $(8, −9)$.

• Find the location of the circumcenter.

Area of Triangles

• Area of triangle trig formula.

• $Area = \frac{1}{2} \cdot side \cdot side \cdot sin(Angle)$

Area Continued

• Given the area, find what’s missing.

• Area is $120 cm^2$

• Find the height and find the missing angle and the missing side.

Honors Geometry Day 1 Chapter 5

• A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle.

• A circle with center P is called “circle P” and can be written as ⊙P.

Lines and Segments That Intersect Circles

• Radius: A segment whose endpoints are the center and any point on a circle.

• Chord: A segment whose endpoints are on a circle.

• Diameter: A chord that contains the center of the circle.

• Secant: A line that intersects a circle in two points.

• Tangent: A line in the plane of a circle that intersects the circle in exactly one point.

• The point of tangency is the point where the tangent line intersects the circle.

Circumference of a Circle

• The circumference of a circle is the distance around the circle.

• The ratio of the circumference C to the diameter d is the same for all circles. This ratio is $\pi$

• Solving for C yields the formula for the circumference of a circle, $C = \pi d$

• Because $d = 2r$, where r is the radius, you can also write the formula as $C = 2\pi r$

• Exact: $2 \pi r$

• Approximate: $2(3.14)r$

Area of a Circle

• The area of a circle is $\pi r^2$, where r is the radius of the circle.

Area of Composite Shapes

• Shape composed of multiple shapes.

• $Area = Area<em>{square} - Area</em>{circle}$

Guided Notes Chapter 6 Day 1

• In a polygon, two vertices that are endpoints of the same side are called consecutive vertices.

• A diagonal of a polygon is a segment that joins two nonconsecutive vertices.

Theorem: Polygon Interior Angles Theorem

• The sum of the measures of the interior angles of a convex n-gon is $(n - 2) \cdot 180°$

EXAMPLE

• Finding the Sum of Angle Measures in a Polygon

• Find the sum of the measures of the interior angles of the figure.

Special Polygons

• In an equilateral polygon, all sides are congruent.

• In an equiangular polygon, all angles in the interior of the polygon are congruent.

• A regular polygon is a convex polygon that is both equilateral and equiangular.

Theorem: Polygon Exterior Angles Theorem

• The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is $360°$

• $m\angle 1 + m\angle 2 + … + m\angle n = 360°$

Summary

• Find the sum of the measures of the interior angles of a convex 15-gon.

• The sum of the measures of the interior angles of a convex polygon is 1620°. Classify the polygon by the number of sides.

Honors Geometry Chapter 8 Similarity

• Corresponding Parts of Similar Polygons

  • A similarity transformation preserves angle measure, so corresponding angles are congruent.

  • A similarity transformation also enlarges or reduces side lengths by a scale factor, so corresponding side lengths are proportional.

Corresponding Lengths in Similar Polygons

• If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons.

Honors Geometry Notes Day 2 Regular Polygon Area

• The center of a regular polygon and the radius of a regular polygon are the center and the radius of its circumscribed circle.

• The distance from the center to any side of a regular polygon is called the apothem of a regular polygon.

• A central angle of a regular polygon is an angle formed by two radii drawn to consecutive vertices of the polygon. To find the measure of each central angle, divide $360°$ by the number of sides of the polygon.

Key Idea: Area of a Regular Polygon

• The area of a regular n-gon with side length s is one-half the product of the apothem a and the perimeter P.

• $Area = \frac{1}{2} aP$

Honors Geometry Ch. 5 Day 2 Circles on the Coordinate plane

• Let $(x, y)$ represent any point on a circle with center at the origin and radius r. By the Pythagorean Theorem, $x^2 + y^2 = r^2$

• This is the equation of a circle with center at the origin and radius r.

Standard Equation of a Circle

• Let $(x, y)$ represent any point on a circle with center $(h, k)$ and radius r. By the Pythagorean Theorem, $(x - h)^2 + (y - k)^2 = r^2$

• This is the standard equation of a circle with center $(h, k)$ and radius r.

Theorem: Angle-Angle (AA) Similarity Theorem

• If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

• If $\angle A \cong \angle D$ and $\angle B \cong \angle E$, then $\triangle ABC \sim \triangle DEF$

Theorem: Side-Side-Side (SSS) Similarity Theorem

• If the corresponding side lengths of the respective triangles are proportional, then the triangles are similar.

• If $\frac{AB}{RS} = \frac{BC}{ST} = \frac{CA}{TR}$, then $\triangle ABC \sim \triangle RST$

Theorem: Side-Angle-Side (SAS) Similarity Theorem

• If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, therefore, the triangles are similar.

• If $\angle X \cong \angle M$ and $\frac{ZY}{PM} = \frac{XY}{MN}$, then $\triangle XYZ \sim \triangle MNP$

Theorems: Triangle Proportionality Theorem

• If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

Theorem: Triangle Angle Bisector Theorem

• If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

Altitudes and Orthocenters

• Altitude: The height of a triangle, connecting a vertex to the opposite side and is perpendicular to the opposite side.

• Orthocenter: The point of concurrency of the altitudes.

  • Located:

  • Inside, on the right angle, or always outside.

What is an angle bisector of a triangle?

• Incenter: Properties of the incenter:

  • Bisects an angle.

  • *Point of Concurrency (POC) of the three angle bisectors.

  • *Always inside the triangle.

  • *Equidistant to the sides of a triangle.

Name ********************** Date*****__ Honors Geometry Chapter 8 Day 3 Theorem: Perimeter of Similar Polygons

• Altitude: The height of a triangle, connecting a vertex to the opposite side and is perpendicular to the opposite side.

• Orthocenter: The point of concurrency of the altitudes.

  • Located:

  • Inside, on the right angle, or always outside.