Geometry (Triangles) Flashcards
Right Triangle
A Triangle with a “right” or 90° angle.
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Hypoteneus
The longest side of a right triangle, opposite the right angle.
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Pythagorean Theorem
a2 + b2 = c2
- In a Right Triangle, the area of the square whose side is the “hypotenuse” is equal to the sum of the areas of the squares of the other two sides.
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Describe Garfields Proof of the Pythagorean Theorem
- Construct a right triangle with Length “B,” Height “A,” and Hypotenuse “C.”
- Create a Congruent Triangle (Stacked on Top) creating a side “A + B” as the height.
- Label the Angles and Prove that the missing Angle = 90°.
- Create an equation using the Area of a Trapezoid,
Height x ((Top Base + Bottom Base)/ 2)
- Create an equation using the Area of the Triangles,
(Base x Height) / 2 and simplify both as a single equation.
Find the “x” value in the triangle below.
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2√3
Find the x value in the triangle below.
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x = 13
Find the x value in the triangle below.
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x = 6
Find the x value in the triangle below.
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x = 8
45°-45°-90° Triangle Equation for the Hypotenuse
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A = B = (√2/2) C
or
(√2) A = (√2) B = C
Find the length of the unknown sides in the triangle shown below
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8√2
Find the length of the unknown sides in the triangle shown below
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5√2
Create a Proof for the 45°-45°-90° Triangle Equation
- Establish that a 45°-45°-90° Triangle has two equal length sides.
- From that information we can Establish that in the Pythagorean Theorem, A = B, and we can Substitute one with the other.
- Simplify the equation and solve for A or B to determine that
A = B = (√2/2) C
or
(√2) A = (√2) B = C
Destinguishing a 45° 45° 90° Triangle
- The two sides that aren’t the hyptotenuse will be the same length
- When given the hypotenuse, the other side of the triangle will be half the length of the Hypotenuse multiplied by √2.
- To find the hypotenuse just multiply the a leg of the Triangle by √2.
Destinguishing the three sides of a 30° 60° 90° Triangle
- The side opposite the 90° angle is equal to “x.”
- The side opposite the 30° angle is 1(x)/2.
- The side opposite the 60° angle is √3(x)/2.
Create a proof for the sides of a 30° 60° 90° Triangle.
- Create an Equilateral Triangle and label all the sides and angles.
- Split the Equilateral Triangle down the middle and label the new sides and angles.
- Create an equation using the Pythagorean Theorem to find the missing side.
- Solve and Simplify the equation for the missing side “a.”