NCM2 Similar Triangles and Special Right Triangles Review

Similar Triangles +
Special Right Triangles
REVIEW

By Kristal Howard

Similar Triangles Review

Dr. Tom Giles

Similar Triangles

Similar Triangles

Difference in Similar & Congruent

​What does similar mean?

• ​Similar shapes are shapes that each side is scaled up by the same amount (proportional) and all angles are congruent/the same

Fill in the Blanks

Fill in the Blanks

​Conditions for similar triangles

• ​AA~

  • ​TWO PAIRS of angles from the triangles are congruent/the same

• ​SAS~

  • ​One angle is congruent to an angle on the other triangle

  • ​The sides on either side of the angle known have same scale factor

• ​SSS~

  • All 3 pairs of sides have same scale factor

Multiple Choice

What is the condition of the similar triangles?

• 3 sides proportional

• ratio of two sides, included $$\angle$$

• A.A.A.

Multiple Choice

Determine whether the triangles are similar by AA~, SSS~, SAS~, or not similar.

• AA

• SSS

• SAS

• Not similar

Multiple Choice

Determine whether the triangles are similar by AA~, SSS~, SAS~, or not similar.

• SAS

• AA

• SSS

• not similar

Multiple Choice

Determine whether the triangles are similar by AA~, SSS~, SAS~, or not similar.

• AA

• SSS

• SAS

• no similar

Multiple Choice

Determine whether the triangles are similar by AA~, SSS~, SAS~, or not similar.

• AA

• SAS

• SSS

• not similar

Multiple Choice

Determine whether the triangles are similar by AA~, SSS~, SAS~, or not similar.

• AA

• SAS

• SSS

• Not Similar

Using Similarity to solve problems

• To find missing sides using similarity, you must first find the scale factor or common ratio between the corresponding sides.

• As you see in the example, all of the sides have a ratio of 1/2, comparing the small triangle to the large triangle

• All of the angles are the same, as is true in all similar figures.

Using Similarity to solve problems

• To find the value for x, the ratio must be the same as the other sides shown.

• The scale factor is 2, so 2x8=16. 16 would be the value for x.

• You try to find the value for y and enter it on the next slide

Fill in the Blanks

Multiple Select

What is true about similar figures

• one figure can be mapped to the other through a series of transformations

• there is a common ratio between corresponding parts

• all angles are the same

• all sides are the same

• a dilation can be used to map one figure to the other

Special Right Triangles

Deriving the Ratios of the Sides

30-60-90 Triangles

30-60-90 Triangle

• Side length are always in these proportions

• We will use this when we are deriving the Unit Circle

Using the ratios to solve triangles

• Determine what you are given: short leg, long leg, or hypotenuse

• Then use the given side and its ratio to find the variable

• Make sure you rationalize the denominator

Multiple Choice

• a=√6   b=√2
• a=4   b=2
• a=√6   b=2
• a=2   b4

Multiple Choice

What are x and y in this 30-60-90 triangle?

• x = 6 y = 3√3
• x = 1.5√3 y =1.5
• x = 3√3 y = 6
• x = √3 y = 2√3

Multiple Choice

What is the measure of side c?

• 6
• 6√2
• 6√3
• 12

Multiple Choice

Find x.

• 10
• 20
• 10√3
• √3

Multiple Choice

Find x.

• 4
• 2
• √3
• 2√2

Multiple Choice

Find x.

• 10
• √3
• 5
• 10√3

Deriving the Ratios of the Sides

45-45-90 Triangles

45-45-90 Triangle

• Side length are always in these proportions

• Congruent angles have congruent sides across from them

• We will use this when deriving the Unit Circle.

Using the ratios to solve triangles

• Determine what you are given: short leg, long leg, or hypotenuse

• Then use the given side and its ratio to find the variable

• Make sure you rationalize the denominator

Multiple Choice

Determine the value of c.

• c = 6√3

• c = 12√3

• c = 18

• c = 18√2

Multiple Choice

Determine the value of x.

• x = 10√2

• x = 10

• x = 5√3

• x = 5√6

Multiple Choice

Use the 45-45-90 theorem to solve for the hypotenuse.

• 16

• 8

• 8√2

• √16

Multiple Choice

Find x - the length of the hypotenuse of the triangle.

• 5
• 5√2
• 10
• 5√3

Multiple Choice

What is the measure of side c?

• 7

• $$\frac{7}{\sqrt{2}}$$

• 7√2

• 14

Multiple Choice

Find x.

• 12
• √2
• √3
• 6

Deriving the Ratios of the Sides

30-60-90 Triangles

30-60-90 Triangle

• Side length are always in these proportions

• We will use this when we are deriving the Unit Circle

Using the ratios to solve triangles

• Determine what you are given: short leg, long leg, or hypotenuse

• Then use the given side and its ratio to find the variable

• Make sure you rationalize the denominator

Multiple Choice

• a=√6   b=√2
• a=4   b=2
• a=√6   b=2
• a=2   b4

Multiple Choice

What are x and y in this 30-60-90 triangle?

• x = 6 y = 3√3
• x = 1.5√3 y =1.5
• x = 3√3 y = 6
• x = √3 y = 2√3

Multiple Choice

What is the measure of side c?

• 6
• 6√2
• 6√3
• 12

Multiple Choice

Find x.

• 10
• 20
• 10√3
• √3

Multiple Choice

Find x.

• 4
• 2
• √3
• 2√2

Multiple Choice

Find x.

• 10
• √3
• 5
• 10√3

Deriving the Ratios of the Sides

45-45-90 Triangles

45-45-90 Triangle

• Side length are always in these proportions

• Congruent angles have congruent sides across from them

• We will use this when deriving the Unit Circle.

Using the ratios to solve triangles

• Determine what you are given: short leg, long leg, or hypotenuse

• Then use the given side and its ratio to find the variable

• Make sure you rationalize the denominator

Multiple Choice

Determine the value of c.

• c = 6√3

• c = 12√3

• c = 18

• c = 18√2

Multiple Choice

Determine the value of x.

• x = 10√2

• x = 10

• x = 5√3

• x = 5√6

Multiple Choice

Use the 45-45-90 theorem to solve for the hypotenuse.

• 16

• 8

• 8√2

• √16

Multiple Choice

Find x - the length of the hypotenuse of the triangle.

• 5
• 5√2
• 10
• 5√3

Multiple Choice

What is the measure of side c?

• 7

• $$\frac{7}{\sqrt{2}}$$

• 7√2

• 14

Multiple Choice

Find x.

• 12
• √2
• √3
• 6

Deriving the Ratios of the Sides

30-60-90 Triangles

30-60-90 Triangle

• Side length are always in these proportions

• We will use this when we are deriving the Unit Circle

Using the ratios to solve triangles

• Determine what you are given: short leg, long leg, or hypotenuse

• Then use the given side and its ratio to find the variable

• Make sure you rationalize the denominator

Multiple Choice

• a=√6   b=√2
• a=4   b=2
• a=√6   b=2
• a=2   b4

Multiple Choice

What are x and y in this 30-60-90 triangle?

• x = 6 y = 3√3
• x = 1.5√3 y =1.5
• x = 3√3 y = 6
• x = √3 y = 2√3

Multiple Choice

What is the measure of side c?

• 6
• 6√2
• 6√3
• 12

Multiple Choice

Find x.

• 10
• 20
• 10√3
• √3

Multiple Choice

Find x.

• 4
• 2
• √3
• 2√2

Multiple Choice

Find x.

• 10
• √3
• 5
• 10√3

Deriving the Ratios of the Sides

45-45-90 Triangles

45-45-90 Triangle

• Side length are always in these proportions

• Congruent angles have congruent sides across from them

• We will use this when deriving the Unit Circle.

Using the ratios to solve triangles

• Determine what you are given: short leg, long leg, or hypotenuse

• Then use the given side and its ratio to find the variable

• Make sure you rationalize the denominator

Multiple Choice

Determine the value of c.

• c = 6√3

• c = 12√3

• c = 18

• c = 18√2

Multiple Choice

Determine the value of x.

• x = 10√2

• x = 10

• x = 5√3

• x = 5√6

Multiple Choice

Use the 45-45-90 theorem to solve for the hypotenuse.

• 16

• 8

• 8√2

• √16

Multiple Choice

Find x - the length of the hypotenuse of the triangle.

• 5
• 5√2
• 10
• 5√3

Multiple Choice

What is the measure of side c?

• 7

• $$\frac{7}{\sqrt{2}}$$

• 7√2

• 14

Multiple Choice

Find x.

• 12
• √2
• √3
• 6