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# Category Archives: Triangles

## 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!

## Vector word problems

## Horizontal and Vertical Components of a Force Vector problem example

A force of 46.3 pounds is applied at an angle of 34.8º to the horizontal. Resolve the force into horizontal and vertical components.

Solution to this **Horizontal Vertical Components of a Force Vector word** practice problem is provided in the video below!

## HARD Parallel Perpendicular Weight Vector Components to Surface word problem

A weight of 75 pounds is resting on a surface inclined at an angle of 25º to the ground. Find the components of the weight parallel and perpendicular to the surface.

Solution to this **Parallel Perpendicular Weight Vector Components word** practice problem is provided in the video below!

## VERY HARD Resultant Force Vector example question

Find the resultant of two forces, one with magnitude 155 pounds and direction *N*50º*W*, and a second with magnitude 305 pounds and direction *S*55º*W*.

Solution to this **Resultant Force Vector word** practice problem is provided in the video below!

## Plane Wind vector example problem

An airplane has an airspeed of 430 miles per hour at a bearing of *E*45º*S* (45 degrees South of East). The wind velocity is 35 miles per hour in the direction of *N*30º*E* (30 degrees East of North). Find the ground speed and true course of the plane using vectors.

Solution to this **Plane Wind vector word** practice problem is provided in the video below!

## Law of Cosines problems

## Law of Cosines word problem example

Two sides of a parallelogram are 9 and 15 units in length. The length of the shorter diagonal of the parallelogram is 14 units. Find the length of the long diagonal.

Solution to this **Law of Cosines word** practice problem is provided in the video below!

## HARD Law of Cosines word problem

A new car leaves an auto transport trailer for a test drive in the flat surface desert in the direction N47ºW at constant speed of 65 miles per hour. The trailer proceeds at constant rate of 50 miles per hour due East. If the car has enough fuel for exactly 3 hours of riding at constant speed, what is the maximum distance in the same direction that the car can cover in order to safely return to the trailer?

Solution to this **Law of Cosines word** practice problem is provided in the video below!

## Law of Sines problems

## Law of Sines word problem example

A radio antenna is attached to the top of the building. From a point 12.5 meters from the base of the building, on level ground, the angle of elevation of the bottom of the antenna is 47.2 degrees and the angle of elevation of the top is 51.8 degrees. Find the height of the antenna.

Solution to this **Law of Sines word** practice problem is provided in the video below!

## Law of Sines Triangle Area PROOF problem

Show that for any triangle the area is one-half the product of any two sides and sine of the angle formed between these sides. That is,

Solution to this **Law of Sines Triangle Area PROOF** practice problem is provided in the video below!

## Inverse Trigonometric Functions problems

## Hard Inverse Trigonometric Expression Value example question

Evaluate the following Trigonometric Expression:

Solution to this **Inverse Trigonometric Expression Value** practice problem is provided in the video below!

## Rectangle Word Problem with ArcTangent example

A rectangle is 173 meters long and 106 meters high. Find the angle between diagonal and the longer side.

Solution to this **Inverse Trigonometric Function word** practice problem is provided in the video below!

## Trigonometric Angle Sum, Difference, Multiple, Half-Angle Formulas problems

## Trigonometric Angle Sum and Difference example question

Find the value of the following Trigonometric Expressions using Angle Sum or Difference formulas:

cos(105º)

Solution to this **Trigonometric Angle Sum Difference** practice problem is provided in the video below!

## Reference Angles problems

## Finding Trigonometric Values Using Reference Angles example question

Find the value of the following Trigonometric Expressions using appropriate Reference Angles:

a) sin(120º)

b)

c) tan(-45º)

d)

e) sec(240º)

f)

Solution to this **Reference Angle** 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!

## Geometric Formula as a Function problems

## Perimeter as a Function of Area example question

a. Express the area, ** A** , of an equilateral triangle as a function of one side

**.**

*S*b. Express the perimeter, ** P** , of the triangle as a function of the area

**.**

*A*

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

## Polar Coordinates problems

## Polar Rectangular Coordinates Conversion example question

Convert the following Rectangular coordinates to Polar coordinates:

a. (2, -2)

b. (0, 3)

c. (3, 4)

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

## Rectangular Polar Coordinates Conversion example problem

Convert the following Polar coordinates to Rectangular coordinates:

a.

b. (0, 3)

c.

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

## Rectangular Polar Coordinates Conversion example

Convert the following Polar coordinates to Rectangular coordinates:

a.

b.

c.

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

## Rectangular Equation & Polar Equation Conversion example question

Find Polar Equations from Rectangular Equation

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

## Polar Equations Sketch & *X*–*Y* Equation Conversion example problem

Sketch the given Polar Equations and find their corresponding *X*–*Y* Equation:

a. *r* = 4

b.

c.

d.

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

## Hard Polar Equations Sketch example

Sketch the given Polar Equations:

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

## Parametric Equation problems

## Parametric Equation Sketch & *X*–*Y* Equation Conversion example question

Sketch the given Parametric Equation and find its corresponding *X*–*Y* Equation:

Solution to this **Parametric Equations & Curves **practice problem is given in the video below!

## Parametric Equation Sketch & *X*–*Y* Equation Conversion example problem

Sketch the given Parametric Equation and find its equivalent *X*–*Y* Equation:

Solution to this **Parametric Equations & Curves **practice problem is given in the video below!

## Parametric Equation Sketch & *X*–*Y* Equation Conversion example

Find Parametric Equations from the given *X*–*Y* equation:

Circle with radius 3, centered at (2, 1) and drawn counterclockwise.

Solution to this **Parametric Equations & Curves **practice problem is given in the video below!

## Parametric Equation Sketch & *X*–*Y* Equation Conversion example question

Derive Parametric Equations from the given *X*–*Y* equation:

Line from (-2, 4) to (6, 1).

Solution to this **Parametric Equations & Curves **practice problem is given in the video below!

## Parametric Equation Sketch & *X*–*Y* Equation Conversion example problem

Find Parametric Equations by using the given *X*–*Y* equation:

from (2, -2) to (0, 2).

Solution to this **Parametric Equations & Curves **practice problem is given in the video below!

## Parametric Equations: Points of Intersection of Two Curves example

Find the coordinates of the points of Intersection of two given Parametric Curves:

Solution to this **Parametric Equations & Curves **practice problem is given in the video below!

## Parametric Equations of a PROJECTILE example problem

A projectile is fired at an angle of inclination *α*, with 0 < *α *< π/2, at an initial speed of *v*_{o}. Parametric equations for its path can be shown to be ** x** =

*v*

_{o}

*t*cos(

*α*),

**=**

*y**v*

_{o}

*t*sin(

*α*) – (

*gt*²)/2, with

*t*representing time.

a) Eliminate the *t* parameter and find the value of *t* when the projectile hits the ground.

b) Sketch the path of the projectile for the case *α* = π/6, *v*_{o} = 32 ft/sec, *g* = 32 ft/sec ².

Solution to this **Parametric Equations of a Projectile **practice problem is given in the video below!

## Parametric Equations: Object Position, Velocity & Speed example question

Find the Position, Velocity and Speed of an Object by using the following Parametric Equation:

Solution to this **Parametric Equations & Curves **practice problem is given in the video below!

## Slope of the Parametric Curve example problem

Find the Slope of the Tangent Line to the Curve

Solution to this **Parametric Equations & Curves **practice problem is given in the video below!

## Parametric Curves: Coordinates of the Points of Vertical & Horizontal Tangent Lines example

Find the Coordinates of the Points of Vertical and Horizontal Tangent Lines to the Curve

Solution to this **Parametric Equations & Curves **practice problem is given in the video below!

## Parametric Curves: Arc Length example question

Find the Arc Length of a Parametric Curve

Solution to this **Parametric Equations & Curves **practice problem is given in the video below!

## Parametric Curves: Area example problem

Find the Area enclosed by a Parametric Curve

in the interval

Solution to this **Parametric Equations & Curves **practice problem is given in the video below!

## Conic Section Proof problems

## Parabola Directrix Conic Section Proof example question

Show that in polar coordinates the equation of a conic section with (one) focus at the pole and directrix,

the line

can be written as

.

Solution to this **Conic Section Proof** practice problem is given in the video below!

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