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# Category Archives: Geometry

## Angular Speed and Linear Speed problems

## Converting between Angular Speed and Linear Speed example question

The angular speed ** ω** of a point moving in a circle is defined as the quotient

**/**

*θ***, where**

*t***is the angle in radians through which the point travels in time**

*θ***.**

*t*a) Find the angular speed of a point that moves through an angle of 4 radians in 6 seconds

b) Find the angular speed of a point on the rim of a wheel that travels at 60 rpm (revolutions per minute)

c) Show that the linear speed ** v** of a point moving in a circle is related to the angular speed by the formula

**=**

*v*

**rω**d) A car is moving at the rate of 60 miles per hour, and the diameter of each wheel is 2.5 feet. Find the angular speed of the wheels.

Solution to this **Angular Speed Linear Speed** practice problem is provided in the video below!

## Geometry Puzzles and Brain Teasers

## Total Number of Triangles Puzzle

How many triangles are there in the figure below?

Solution to this **Geometry** **Triangle** **Puzzle** problem is given in the video below!

## Total Number of Rectangles Puzzle

How many rectangles are there in the figure below?

Solution to this **Geometry** **Rectangle** **Puzzle** is given in the video below!

## Total Number of Upright SQUARES in a Square

How many upright **Squares** are there in the following square?

The key is to understand vertices of any given upright square and a diagonal line of the largest square to see how many translations of that given upright square you can have. As such, there is a summation-derived FORMULA you can actually use for any largest size square to find the total number of possible upright squares contained within it!

Solution to this **Geometry** **Square** **Puzzle** is given in the video below!

## Total Number of Upright RECTANGLES in a Square (or Rectangle)

How many upright **Rectangles** are there in the following square?

The key is to understand vertices of any given upright rectangle to see how many unique translations of that rectangle are possible. As such, there is actually a summation-derived FORMULA you can utilize for the largest square or rectangle of any size to find the total number of possible upright rectangles within it!

Solution to this **Geometry** **Rectangle** **Puzzle** is given in the video below!

## Total Number of Upright SQUARES in a Rectangle

How many upright **Squares** are there in the following rectangle?

The key is to understand vertices of any given upright square to see how many translations of that given square you can have. As such, there is a summation-related FORMULA you can actually derive for any size rectangle to find the total number of possible upright squares contained within it!

Solution to this **Square** **Rectangle** **Geometry** **Puzzle** is given in the video below!

## Total Number of Upright & Tilted SQUARES in a Square

How many upright and tilted **Squares** are there in the following square?

The key is to use vertices of any given largest square to see how many translations of a given smaller square you can have. As such, there is a summation-derived FORMULA you can actually use for any size square to find the total number of possible upright and tilted squares contained within it!

Solution to this **Square** **Geometry** **Puzzle** example is given in the video below!

## Total Number of Upright & Tilted SQUARES in a Rectangle

How many upright and tilted **Squares** are there in the following rectangle?

The key is to use vertices of any given rectangle to see how many translations of a given square you can have. As such, there is a summation-derived FORMULA you can actually use for any size rectangle to find the total number of possible upright and tilted squares contained within it!

Solution to this **Rectangle** **Geometry** **Puzzle** example problem is given in the video below!

## Translation Geometry problems

## Translation Coordinates and Graph of Translated Image example question

Find the coordinates and the translated graph of **A**(-2, 3), **B**(4, -5), and **C**(0, 3) by following the rule of translation:

(*x*, *y*) → (*x* – 5, *y* + 6).

Solution to this **Translation** Geometry practice problem is given in the video below!

## Finding Coordinate RULE of Translation given Graph of Translated Image example problem

Find the coordinate rule for the translation shown in the graph below.

Solution to this **Translation** Geometry practice problem is given in the video below!

## Dilation Geometry problems

## Dilation Formulas and Graph with Origin the Center of Dilation example question

Graph the triangle with vertices **A**(-2, 4), **B**(1, -4), **C**(-3, -2) and its image after dilation by scale factor ** k** = -3. The center of dilation is the origin.

Solution to this **Dilation** Geometry practice problem is given in the video below!

## Dilation Formulas and Graph with Center of Dilation NOT the Origin example problem

Graph the triangle with vertices **A**(-2, 4), **B**(1, -4), **C**(-3, -2) and its image after dilation by scale factor ** k** = 2. The center of dilation is the point (4, 5).

Solution to this **Dilation** Geometry practice problem is given in the video below!

## How to Find the CENTER of DILATION COORDINATES using Given Figure and its Image problem

Find the coordinates of the center of dilation if coordinates of original figure are (-4, 14), (-10, 16), (-12, 10), (-14, 12) and coordinates of its image are (2, 1), (3, 4), (-1, 2), (4, 3).

Solution to this **Dilation** Geometry practice example is given in the video below!

## Reflection Geometry problems

## Reflection Formulas example question

Draw the image of the polygon with coordinates **A**(-2, 3), **B**(0, 2), **C**(3, -4), **D**(4, 0) by reflecting it

a) through the ** y**-axis

b) through the ** x**-axis

c) through the **origin **

d) through the line *y* = *x*

Solution to this **Reflection** Geometry practice problem is given in the video below!

## Rotation Geometry problems

## Rotation Formulas example question

a) Draw the image of the triangle with coordinates **A**(-6, -6), **B**(-6, 3), **C**(-2, 3) by rotating it 90 degrees clockwise

b) Draw the image of the quadrilateral with coordinates **A**(-3, -3), **B**(-1, 0), **C**(3, 0), **D**(5, -3) by rotating it 90 degrees **counter**clockwise

c) Draw the image of the triangle with coordinates **A**(-3, 2), **B**(1, 5), **C**(0, 0) by rotating it 90 degrees clockwise

d) Draw the image of the triangle with coordinates **A**(1, -3), **B**(3, 3), **C**(6, -3) by rotating it 180 degrees clockwise or **counter**clockwise

e) Draw the image of the quadrilateral with coordinates **A**(-5, -2), **B**(-4, 5), **C**(2, 5), **D**(2, 0) by rotating it 90 degrees clockwise

Solution to this **Rotation** Geometry practice problem is given in the video below!

## Solving for x in Angles and Triangles problems

## Solving for *X* and *Y* in Angles example question

Find the values of *x* and *y* using the figure below.

Use these values to find the following:

a)

b)

c)

d)

Solution to this **value of x and y in Angles **Geometry practice problem is provided in the video below!

## Angles, Parallel Lines and Transversals problems

## Angle Theorems Parallel Lines and Transversals example question

The following figure shows two parallel lines *l* and *m* and a transversal.

Use this figure to answer the following questions:

a)

b)

c)

d)

e)

f)

g)

Solution to this **Angles Parallel Lines Transversal **Geometry practice problem is provided in the video below!

## Angle Theorems Parallel Lines and Transversals example problem #2

Use the following figure to find the values of *x *and *y*.

Solution to this **Angles Parallel Lines Transversal **Geometry practice problem is provided in the video below!

## Angle Theorems Parallel Lines and Transversals example #3

Use the figure below to solve for the values of *x *and *y*.

Solution to this **Angles Parallel Lines Transversal **Geometry practice problem is provided in the video below!

## Angle Theorems Parallel Lines and Transversals example question #4

Use the given figure below to find the values of *x *and *y*.

**Angles Parallel Lines Transversal **Geometry practice problem is provided in the video below!

## Angle Theorems Parallel Lines and Transversals example problem #5

The figure below shows a polygon with some interior angle measures provided.

Find the values of *x *and *y*.

**Angles Parallel Lines Transversal **Geometry practice problem is provided in the video below!

## World’s HARDEST Easy Geometry problems

## World’s HARDEST Easy Geometry problem example question

The following triangle, , shows angle measures in the base angles ** A **and

**, as well as line segments , and . Find the measure of angle**

*C**AED*, that is, .

**Note:** Do not use Trigonometry or Calculus to solve this problem

Solution to this **World’s Hardest Easy Geometry** practice problem is provided in the video below!

## Orthocenter Outside the Obtuse Triangle problems

## Centroid Circumcenter Incenter Orthocenter properties example question

In the following video you will learn how to find the coordinates of the Orthocenter located outside the triangle in the standard *xy*-plane (also known as coordinate plane or Cartesian plane). In acute and right triangles, the Orthocenter does not fall outside of the triangle. However, when the triangle in question is **obtuse**, that is, when one of its interior angles measures more than 90 degrees – the Orthocenter will be located outside the triangle. This is when you will need to understand the technique used to find its coordinates.

Solution to this **Orthocenter in Obtuse Triangle **Geometry practice problem is provided in the video below!

## Centroid Circumcenter Incenter Orthocenter problems

## Centroid Circumcenter Incenter Orthocenter properties example question

In this video you will learn the basic properties of triangles containing Centroid, Orthocenter, Circumcenter, and Incenter. Then you can apply these properties when solving many algebraic problems dealing with these triangle shape combinations.

Solution to this **Centroid Circumcenter Incenter Orthocenter **Geometry practice problem is provided in the video below!

## Two Column Geometry Proof problems

**Two Column Geometry Proof Intersecting Lines example problem**

**Given: **GA = RA, AH = AT

**Prove:** GT is congruent to RH

Solution to this **Two Column Geometric Proof** practice problem is given in the video below!

**Two Column Geometry Proof Trapezoid example**

**Given:** AD = 8, BC = 8, and BC is congruent to CD

**Prove:** AD is congruent to CD

Solution to this **Two Column Geometric Proof** practice problem is given in the video below!

**Two Column Geometry Proof Circle example question**

**Given:** is a diameter

**Prove:** (*CD*)^{2} = (*AD*)(*DB*)

Solution to this **Two Column Geometric Proof** practice problem is given in the video below!

**Two Column Geometry Proof Triangle example problem**

**Given:**

**Prove:**

Solution to this **Two Column Geometric Proof** practice problem is given in the video below!

**Two Column Geometry Proof Triangle example #2**

**Given:** with the bisector of

**Prove:** *BFDE* is a rhombus

Solution to this **Two Column Geometric Proof** practice problem is given in the video below!

**Two Column Geometry Proof Circle example question #2**

**Given: **Circle with center *O* and diameter

**Prove: **

Solution to this **Two Column Geometric Proof** practice problem is given in the video below!

**Two Column Geometry Proof Kite example problem**

**Given:** Kite *ABCD* with

*E*, *F*, *G*, *H* are midpoints of

**Prove:** *EFGH* is a rectangle

Solution to this **Two Column Geometric Proof** practice problem is given in the video below!

**Two Column Geometry Proof Triangle example #3**

**Given:** bisects

is a perpendicular bisector of

**Prove:**

Solution to this **Two Column Geometric Proof** practice problem is given in the video below!

**Two Column Geometry Proof Triangle example question #4**

**Given: ** is isosceles and

**Prove:**

Solution to this **Two Column Geometric Proof** practice problem is given in the video below!

**Two Column Geometry Proof Parallel Lines Transversal Triangle example problem**

**Given:**

bisects

bisects

**Prove:**

Solution to this **Two Column Geometric Proof** practice problem is given in the video below!

## Shaded Area in Geometric Figure puzzles

## Shaded Area in a Square puzzle example question

In a unit square *ABCD*, point *A* is joined to the midpoint of *BC*, point *B* is joined to the midpoint of *CD*, point *C* is joined to the midpoint of *DA*, and point *D* is joined to the midpoint of *AB*. Find the area of the shaded region.

Solution to this **Puzzle** practice problem is given in the video below!

## Shaded Area in a Square puzzle example problem

In a unit square *ABCD*, *M* is a midpoint of *AD*, and *AC* is a diagonal. Find the area of the shaded regions.

Solution to this **Puzzle** practice problem is given in the video below!

## Shaded Area in a Square puzzle example

In the unit square *ABCD*, *M* is the midpoint and *AC* and *BD* are diagonals. Find the area of the shaded region.

Solution to this **Puzzle** practice problem is given in the video below!

## Shaded Area in a Square puzzle example question

In the following unit square *ABCD*, *M* is the midpoint and *BD* is a diagonal. Find the area of the shaded region.

Solution to this **Puzzle** practice problem is given in the video below!

## Shaded Area in a Square puzzle TRIGONOMETRY example problem

A square with side 1 is rotated around one vertex by an angle , where

and

.

Find the area of the shaded region.

Solution to this **Puzzle** practice problem is given in the video below!

## Shaded Area in a Rectangle puzzle example

In the following rectangle, an isosceles triangle is drawn. Find the area of the shaded region.

Solution to this **Puzzle** practice problem is given in the video below!

## Shaded Area in a Rectangle puzzle example question

In the following rectangle below, find the area of the shaded region.

Solution to this **Puzzle** practice problem is given in the video below!

## Geometry problems (complete Playlist)

Explore a variety of **Geometry** examples and practice problems applicable to K-12, Precalculus or Calculus courses.

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