Common Core: High School - Geometry Help » Similarity, Right Triangles, & Trigonometry » 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

### Example Question #1 : Derive The Formula A = một nửa Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular to lớn The Opposite Side

Is the following statement True or False?

We want khổng lồ use the formula . Consider an obtuse triangle . We know the lengths of and , but only know the angle for . We are still able to lớn use this formula.

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Explanation:

There are two approaches to this problem. We are able to calculate the angle by using the Sine Law. The Sine Law states: So we can mix and solve accordingly for angle .

Our other option is to lớn use the area formula we have been but altering it to correspond khổng lồ angle . We would draw our vertical line down from the vertex as shown below và our formula would be in the size . ### Example Question #2 : Derive The Formula A = một nửa Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular khổng lồ The Opposite Side

Solve for x using the formula given that the area of the following triangle is (round lớn the second decimal place if needed).   Explanation:

Even though the formula is using sides 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 bởi some more work to get the súc tích right.

Xem thêm: Đặt Câu Với Từ Học Tập Lớp 2, Soạn Bài Luyện Từ Và Câu: Từ Ngữ Về Học Tập Using the figure above, lớn be able lớn 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 đứng đầu vertex to size a right angle. This vertical line is . Angle for the supplementary (orange) triangle is . Using the fact that , we can mix up our formula to be the following:   (either angle can be used and I will show this to lớn be true)  (Using from the original triangle)  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 lớn draw this supplementary triangle in order to lớn be able to visualize and understand logically how the formula is working for obtuse angles as well.