Heron s Formula. Key Words: Triangle, area, Heron s formula, angle bisectors, incenter


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1 Heron s Formula Lesson Summary: Students will investigate the Heron s formula for finding the area of a triangle. The lab has students find the area using three different methods: Heron s, the basic formula, and then using Cabri. Constructions for proving Heron s theorem are included as are two extensions and a practice worksheet. Key Words: Triangle, area, Heron s formula, angle bisectors, incenter Existing Knowledge: Students should be familiar with the following concepts: area of triangles and angle bisectors. Learning Objectives:. Students will be able to construct the incenter of a triangle.. Students will be able to make conjectures and describe relationships between the area of triangles using Heron s theorem and the traditional formula. Materials:. Laboratory worksheet. Access to computer lab or calculator equipped with Cabri Geometry II. Procedure: Split the class into groups of two or three. Have them complete the worksheet and extensions. Assessment: Choices for the types of assessment are left to the discretion of the instructor. Some examples might include: the completed lab questions, class participation (student explanations of the lab, conjectures, answers and/or processes in finding solutions to the extension problems or student created extensions), and peer or selfevaluation. Keep in mind that several forms of assessment should be utilized when lessons are inquiry/discovery based.
2 Heron s Formula Team Members names: File name: Goal: Investigate alternative ways to compute the area of a triangle, given the three side lengths and to verify that Heron s Formula works. Investigation Create ABC using Cabri Geometry II. Make it fill up most of the screen. (triangle and label tools) Now, we will have Cabri help us find the area of a triangle three different ways. The first way takes a while to develop, but the next two get much easier. Method A Finding the incenter to develop part of Heron s Formula.. Construct the angle bisector of A, B and C, make them dashed lines and label their intersection point P. (angle bisetor, dashed and label tools). Create perpendicular lines from point P to sides AB, AC and BC and label the points X, Y and Z respectively as shown below. (perpendicular line and label tool)
3 3. Construct a circle with center P and this radius point being X, Y or Z. (circle tool) 4. Grab point A, B or C with the pointer and move it around until you re convinced the circle is truly inscribed in the triangle. 5. Now, on your drawing, hide all six lines (3 perpendicular and 3 angle bisectors) and construct segments AP, BP and CP. (hide/show and segment tools) 6. Call the three original sides of the triangle a, b and c (opposite sides A, B and C respectively). (comments tool) 7. The area of ABC is equal to the sum of which three smaller triangles? 8. Remember the typical formula for finding the area of a triangle (A = bh). Considering the three original sides of ABC as the base for each of the triangles you listed above for #7, what do you notice about the height of each of these smaller triangles? What do we call this height in relation to the circle?
4 9. Therefore, the area of one of these small triangles is A = (a)(r). Write down a comparable expression for the other two small triangles. 0. Write out the sum of these three triangles using the variables a, b, c and r. Then factor out a r.. What is the sum inside the parentheses commonly know as?. We can now write the above sum as rp, then rearrange to p r, then simplify to sr, by using the substitution s = p, where s is the semiperimeter (half the perimeter). 3. Now, with a little work (which we are not going to do here), we can arrive at Heron s Formula for finding the area of a triangle given the 3 side lengths a, b and c as: a + b + c A = s( s a)( s b)( s c), where s equals, the semiperimeter. Note: A great geometric proof of Heron s Formula is given on pages of David C. Kay s textbook, entitled College Geometry: A Discovery Approach (994). 4. Restate Heron s Formula using only words. 5. Have Cabri find AB, AC and BC and compute the area of your triangle using Heron s Formula and record and label your answer. (measure, calculate and label tools) Next, we re going to find the area of your triangle two other ways, in order to investigate whether Heron s Formula works for all triangles. Method B Finding the altitude to a side and use the formula A = bh. Hide everything in your Cabri drawing except the original triangle, the labeled points A, B and C and the side measurements of the triangle. (hide tool). Make a line that coincides with segment BC. (line tool)
5 . From vertex A, draw the altitude to the line containing the BC and name the intersection point D. (perpendicular line, intersection points and label tools) 3. Find AD. (measure tool) 4. Have Cabri compute the area of your triangle using A = bh and record and label your answer in the worksheet. (calculate and label tools) Method C Let Cabri find the area of your triangle by itself.. Find the area of the triangle. Record it here: (area tool) A Few Questions. Do all three methods for computing the area of your triangle produce the same answer?. Do you think Heron s Formula will work for all acute triangles? All obtuse triangles? All right triangles? 3. To test your conjecture from question #, grab one of the original three vertices and move it wherever you want. Do the three area computations all stay the same as each other? 4. Do you know why the formula A = bh works? Let s check it out. Below, draw and label a rectangle whose length is b and width is h. Show the proper formula, work and answer for the area of this rectangle. Now, draw a diagonal of this rectangle and show the proper formula, work and answer for one of the triangles created by the diagonal. How do the two areas you just found relate to one another?
6 Area of Triangles Practice Worksheet On the following two pages, you will find a worksheet designed to give you some practice finding areas of triangles using Heron s Formula and the traditional formula Heron s Formula A H = s( s a)( s b)( s c) vs. Traditional Formula A T = bh For #5, find the area of each triangle using both of the above formulas. For each problem, show your work on the back of this sheet and also record them in the table below. Show all units. A H A T # # #3 #4 #5.. Find the area. Find x, then find the area. 3. Draw an isosceles right triangle below 4. and label the hypotenuse 8. Find the two legs, then find the area. Find the area. 5. Find the area of the obtuse triangle at the right, where h is approximately Round your answers to the nearest tenth. h 6. Landscaping Problem Tom finds 3 old boards in his garage that he plans to use to make a triangular flowerbed. The first board is 6 feet long, the second one is 9 feet long and the third is feet long. He has no saw, no tape measure and no protractor. How many bags of mulch (each is.5 cubic feet) does Tom need to buy if he wants 3 inches of mulch inside the flowerbed? Show and label all work.
7 Extension. Have Cabri show you the coordinates of points A, B and C of your original triangle. (equations and coordinates tool). Let A = (x, y ), B = (x, y ) and C = (x 3, y 3 ). 3. Define the following 3x3 matrix: x x x 3 y y y 3 4. Find the determinant of this matrix. For help, see the following web address or use a matrixcapable calculator. 5. How does this compare to the area of your triangle? 6. What is the general result from the work above?
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