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Flashcards: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

Solve for x using the formula $A = \frac{1}{2}ab(sin(C))$ given that the area of the following triangle is 20cm^2 (round to the second decimal place if needed).

x = 21.14cm

x = 23.09cm

x = 25.11cm

x = 14.15cm

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Flashcards: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

Solve for x using the formula $A = \frac{1}{2}ab(sin(C))$ given that the area of the following triangle is 20cm^2 (round to the second decimal place if needed).

Your answer:

x = 21.14cm

Your answer:

x = 25.11cm

Your answer:

x = 14.15cm

Correct answer:

x = 23.09cm

Explanation:

Even though the formula $A = \frac{1}{2}ab(sin(C))$ is using sides a, b and angle , this is a general formula and can be used with any angle in the triangle. Since we are now working with an obtuse angle rather than an acute angle, we need to do some more work to get the logic right.

Using the figure above, to be able to label the sides we are using correctly, we extend the original triangle horizontally past the obtuse angle and draw a vertical line down from the top vertex to form a right angle. This vertical line is . Angle for the supplementary (orange) triangle is 180 - 120 = 60 . Using the fact that $sin(A) = \frac{h}{b} = \frac{h}{2x}$ , we can set up our formula to be the following:

a = x

b = 2x

C = A (either angle can be used and I will show this to be true)

$A = \frac{1}{2}ab(sin(C))$

$20 = \frac{1}{2}(x)(2x)sin(60)$ (Using from the original triangle)

$\frac{20}{sin(60)} = x^2$

4.81 = x

This shows that when using this formula for an obtuse angle, you can use either the supplementary angle you made, or the original. It is always helpful to draw this supplementary triangle in order to be able to visualize and understand logically how the formula is working for obtuse angles as well.

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All Common Core: High School - Geometry Resources

Flashcards: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

Flashcards: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

Flashcards: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

Flashcards: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

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