Special Right Triangles

Keystone Geometry

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Review: Parts of a Right Triangle 2

Special Types of Right Triangles

A right triangle must have exactly one 90 degree angle.

That leaves the two remaining angles to be acute and complementary.

One type is 45º-45º-90ºAnother type is 30º-60º-90º 3

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* A 45º ｰ 45º ｰ 90º triangle is an isosceles triangle with congruent legs. If the length of a leg is a, then the length of the hypotenuse is a times the square root of 2.

45º- 45º- 90º Special Right Triangle

• In a triangle 45º- 45º- 90º, the hypotenuse is times as long as a leg.

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45°

45°

Hypotenuse

xLeg

Leg

Example:

45º

45°

5 cm

5 cm

5 cm

x

45º- 45º- 90º Special Right Triangle

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* In a 30º ｰ 60º ｰ 90º triangle, the shorter leg is opposite the 30º angle and the longer leg is opposite the 60º angle. The theorem says if the shorter leg has length a, then the hypotenuse has length 2a and the longer leg has length a times the square root of 3.

30º- 60º- 90º Special Right Triangle

• In a triangle 30º- 60º- 90º, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.

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30º

60º

Hypotenuse

3x

2x

x

Longer Leg

Shorter Leg

Example:

30º

60°

10 cm

5 cm

5 cm

30º- 60º- 90º Special Right Triangle

Example: Find the value of a and b.

8

60º7 cm

a

b

Step 1: Find the missing angle measure.

30°

30º°

Step 2: Decide which special right triangle applies.

30º- 60º- 90º

Step 3: Match the 30º- 60º- 90º pattern with the problem.

30º

60ºx

2x

a = cm

b = 14 cm

Step 5: Solve for a and b

Step 4: From the pattern, we know that x = 7, b = 2x and a = x

Example: Find the value of a and b.

2x

9

45º7 cm

a

b

Step 1: Find the missing angle measure.45°

45 °

Step 2: Decide which special right triangle applies.

45º- 45º- 90º

Step 3: Match the 45º - 45º- 90º pattern with the problem.

45º

45ºx

x

Step 4: From the pattern, we know that x = 7 , a = x, and b = x .

a = 7cm

b = 7 cm

Step 5: Solve for a and b