Unit 7 Lesson 7.3: Special Right Triangles

Unit 7 Lesson 7.3:

Special Right

Triangles

MT: Solving Applied Problems & Modeling in Geometry

ISOSCELES RIGHT TRIANGLES

​The "45°-45°-90°" Triangle

In this triangle, the following must be true:

1. Both legs must be CONGRUENT.

2. Both acute angles must be CONGRUENT (45°​).

3. The hypotenuse will always be LARGER than the legs.​

​Leg₁

​Leg₂

Hypotenuse

45°

45°

ISOSCELES RIGHT TRIANGLES

Since the 45°-45°-90° triangle is special, it comes with something called a CONSTANT RATIO.

​

This ratio represents EVERY 45°-45°-90° triangle IN THE WORLD.

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Ratio:

x : x : x√2

leg₁ leg₂ hypotenuse​

ISOSCELES RIGHT TRIANGLES

x : x : x√2

leg₁ leg₂ hypotenuse

​

That "x" can represent ANY NUMBER IN THE WORLD as long as it isn't:

• zero

• negative​

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ISOSCELES RIGHT TRIANGLES

Follow these steps to find the missing sides​:

​

STEP 1: Write down the constant ratio. ​

STEP 2: Write the value of given side UNDERNEATH its constant ratio.

STEP 3: Use the patterns/shortcuts to fill in the other sides.

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Patterns and shortcuts are given during the next example.​

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ISOSCELES RIGHT TRIANGLES

Example:

​Given the triangle to the right, find the value of the legs.

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STEP 1: Write down the constant ratio.

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ISOSCELES RIGHT TRIANGLES

Example:

​Given the triangle to the right, find the value of the legs.

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STEP 2: Write down given value underneath its constant ratio.

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ISOSCELES RIGHT TRIANGLES

Example:

​Given the triangle to the right, find the value of the legs.

​​

STEP 3: Use patterns/shortcuts to fill in other missing sides.

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​

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ISOSCELES RIGHT TRIANGLES

SHORTCUTS: If given a leg FIRST, follow these steps to find the hypotenuse:

1. Add a √​2 to the end of the leg's value

2. Simplify if necessary. ​

​​

SHORTCUTS: If given the hypotenuse FIRST, follow these steps to find the legs:

1. Divide the hypotenuse by 2.

2. Add a √​2 to the end of the value.

3. Simplify if necessary. ​

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FACTORING ROOTS

Sometimes a root can be simplified MORE than it already is. In other words, sometimes theres a root within a root.

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To simplify a root, you FACTOR the value with hopes of finding a PERFECT ROOT (a factor that will simplify to a whole number.

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Follow these steps:

1. Find all factor pairs of the value.

2. Cross out any factor pair including the number 1.

3. Find a pair where at least ONE is the perfect root.

4. Rewrite the original root.

5. Simplify if necessary.

FACTORING ROOTS

Follow these steps:

1. Find all factor pairs of the value.

2. Cross out any factor pair including the number 1.

3. Find a pair where at least ONE is the perfect root.

4. Rewrite the original root.

5. Simplify.

Multiple Choice

Determine the length of the hypotenuse for the given 45-45-90 triangle:

• $$6\ \sqrt[]{2}$$  

• $$2\ \sqrt[]{2}$$  

• $$4\ \sqrt[]{3}$$  

• $$3\ \sqrt[]{2}$$  

Multiple Choice

Determine the length of the legs for the given 45-45-90 triangle:

•   $$16\ \sqrt[]{6}$$  

•   $$32\ \sqrt[]{6}$$  

• $$16\ \sqrt[]{2}$$    

• $$32\ \sqrt[]{2}$$  

Multiple Choice

Determine the length of the hypotenuse for the given 45-45-90 triangle:

•   $$4\ \sqrt[]{5}$$  

•   $$8\ \sqrt[]{5}$$  

•   $$8\ \sqrt[]{10}$$  

• $$8\ \sqrt[]{3}$$    

Multiple Choice

Determine the length of the legs for the given 45-45-90 triangle:

• $$\frac{13\ \sqrt[]{2}}{2}$$     

• $$26\ \sqrt[]{10}$$   

• $$13\ \sqrt[]{10}$$  

• $$\frac{13\ \sqrt[]{10}}{2}$$  

HALF OF AN EQUILATERAL TRIANGLE

​The "30°-60°-90°" Triangle

In this triangle, the following must be true:

1. The legs can NEVER be congruent.

2. The long leg is also referred to as the ALTITUDE.

3. The hypotenuse will always be LARGER than the legs.​

​Leg 1

​Leg 2

Hypotenuse

45°

45°

HALF OF AN EQUILATERAL TRIANGLE

Since the 30°-60°-90° triangle is special, it comes with something called a CONSTANT RATIO.

​

This ratio represents EVERY 30°-60°-90° triangle IN THE WORLD.

​

Ratio:

x : x√3 : 2x

short leg long leg hypotenuse​

HALF OF AN EQUILATERAL TRIANGLE

x : x√3 : 2x

short leg long leg hypotenuse

​

That "x" can represent ANY NUMBER IN THE WORLD as long as it isn't:

• zero

• negative​

The long leg is also known as:

• altitude

• height​

Follow these steps to find the missing sides​:

​

STEP 1: Write down the constant ratio. ​

STEP 2: Write the value of given side UNDERNEATH its constant ratio.

STEP 3: Use the patterns/shortcuts to fill in the other sides.

​

Patterns and shortcuts are given during the next example.​

​

​

HALF OF AN EQUILATERAL TRIANGLE

HALF OF AN EQUILATERAL TRIANGLE

Example:

​Given the triangle to the right, find the value of the long leg & hypotenuse.

​

STEP 1: Write down the constant ratio.

​​

HALF OF AN EQUILATERAL TRIANGLE

Example:

​Given the triangle to the right, find the value of the long leg & hypotenuse.

​

STEP 2: Write the value of given side UNDERNEATH its constant ratio.

​​

HALF OF AN EQUILATERAL TRIANGLE

Example:

​Given the triangle to the right, find the value of the long leg & hypotenuse.

​

STEP 3: Use patterns/shortcuts to fill in other missing sides.

​​

HALF OF AN EQUILATERAL TRIANGLE

SHORTCUTS: If given the short leg FIRST, follow these steps to find the others:

1. Add a √​3 to the end of the short leg's value to find the long leg. Simplify if necessary.

2. Multiply the short leg by 2 to find the hypotenuse.

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HALF OF AN EQUILATERAL TRIANGLE

SHORTCUTS: If given the long leg FIRST, follow these steps to find the others:

1. Divide the long leg by 3, then add another √3 to the end to find the short leg. Simplify if necessary.

2. Multiply the short leg by 2 to find the hypotenuse.

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HALF OF AN EQUILATERAL TRIANGLE

SHORTCUTS: If given the hypotenuse FIRST, follow these steps to find the legs:

1. Divide the hypotenuse by 2 to find the short leg.

2. Add a √​3 to the end of the short leg's value to find the long leg. Simplify if necessary.

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Multiple Choice

Determine the length of the long leg & the hypotenuse for the given 30-60-90 triangle:

• LL = $$14\ \sqrt[]{2}$$  

  HYP = $$14\ \sqrt[]{3}$$  

• LL = $$14\ \sqrt[]{3}$$  

  HYP = $$14\ \sqrt[]{2}$$  

• LL = $$14\ \sqrt[]{3}$$  

  HYP = $$28$$  

• LL = $$14\ \sqrt[]{2}$$  

  HYP = $$28$$  

Multiple Choice

Determine the length of the short leg & the hypotenuse for the given 30-60-90 triangle:

• SL = $$9$$  

  HYP = $$18$$  

• SL = $$27$$  

  HYP = $$54$$  

• SL = $$3\ \sqrt[]{3}$$  

  HYP = $$6\ \sqrt[]{3}$$  

• SL = $$9\ \sqrt[]{6}$$  

  HYP = $$18\ \sqrt[]{6}$$  

Multiple Choice

Determine the length of the short leg & the long leg for the given 30-60-90 triangle:

• SL = $$4\ \sqrt[]{6}$$  

  LL = $$12\ \sqrt[]{2}$$  

• SL = $$8$$  

  LL = $$8\ \sqrt[]{3}$$  

• SL = $$8\ \sqrt[]{2}$$  

  LL = $$8\ \sqrt[]{6}$$  

• SL = $$2\ \sqrt[]{2}$$  

  LL = $$2\ \sqrt[]{6}$$