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Home , Triangle

©Zvezdelina Stankova, Berkeley Math Director, July 23, 2014

9. of a Triangle

Problem 1. Draw a on your graph C paper. Calculate its area. Describe a general

formula for the area of a rectangle.

The area of a rectangle is calculated by:

______A B

D C Problem 3. Complete the ABC above to a rectangle: use the as the of the rectangle and draw some to the legs make the rectangle. What part (how much) A B of the rectangle is the triangle? 1. The area of a rectangle is the product of two adjacent sides: The area of the right triangle ABC is equals :

Area of a rectangle = x width ______= x height.

Lemma 1. The area of a right triangle is half

Problem 2. Now draw a right triangle on of the product of the two legs:

your graph paper? How much is its area? Area of a right triangle equals: Conjecture a formula that should work for (leg x leg) = base x height). any right triangle.

Hint 1. Can you somehow reduce to the Problem 4. Draw on your graph paper an previous situation with the rectangle? Isn’t acute triangle ABC (let AB be horizontal).

every right triangle part of a rectangle? Calculate the area of the triangle ABC.

Definition 1. In a right triangle ABC the Hint 2. Reduce to the previous situation two sides that make the right are with the right : drop the called the legs, and the side opposite the CH from C to side AB. How many right is called the hypotenuse of the triangles do you see? Color them differently.

triangle.

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©Zvezdelina Stankova, Berkeley Math Circle Director, July 23, 2014

Hint 3. Complete the right triangles AHC Definition 2. Let CH be the altitude in and BHC to rectangles. Do you see a big triangle ABC. H is called the foot of rectangle in which the original triangle the altitude, or the foot of the ABC is inscribed? What is the area of this from C to AB. big rectangle? What part (how much) of the

big rectangle is triangle ABC? D C

E C D

A B H

Proof. We already know how to find the

of right triangles AHC and BHC: A H B multiply their legs and divide by 2:

Area of right AHC = AH x HC

Proof. The acute triangle ABC is half of the Area of right BHC = BH x HC rectangle ABDE. Since the base of the rectangle is the side AB and the height of the Since the area of triangle ABC is the rectangle is the altitude CH, the area of the difference of these areas, we obtain: Area of obtuse BHC = rectangle is AB x CH. Hence the area of the triangle is half of that. = AH x CH - BH x CH =

= (AH-BH) x CH= AB x CH. Lemma 2. The area of an acute triangle But this is again half of the base times the

ABC is half of its base x height: height? We are getting the same formula

Area of an acute triangle equals: for any type of triangle!

side x altitude) = (base x height).

Theorem 2. The area of any triangle ABC is

Problem 5. Final challenge: draw on your half of its base x height: Area of any triangle equals: graph paper an obtuse triangle ABC with side AB horizontal and an obtuse angle at B. side x altitude) = (base x height). Drop the height CH from C to () AB.

Complete the right triangle AHC to a

rectangle AHCD. Can you use this picture Question 1. There is one formula for the to find the area of the original triangle area of a rectangle. How many formulas

ABC? Write down your conjecture. are there for the area of a triangle?

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©Zvezdelina Stankova, Berkeley Math Circle Director, July 23, 2014

C Historical Facts. The computation of areas has a long history. The ancient Egyptians, Babylonians, and Hindus knew how to b a compute areas of simple geometric figures h h a b like triangles, , rectangles,

hc , and . In 499

CE , a great mathematician- A c B astronomer from the classical age of Indian

Standard Notation. Denote each side by mathematics and Indian astronomy, used the the lower-case letter of the opposite method of multiplying the base times the to that side: AB = c, BC = a, and CA = b, height and dividing by 2 in the Aryanhatiya.

Denote also the altitude from a vertex by h The Aryanhatiya is a Sanskrit astronomical with subscript the vertex’s lower-case letter: treatise (written in verses!), is the magnum

the altitude from B to side AC is hb, opus and only surviving work of Āryabhaṭa. the altitude from A to side BC is h a, Statue of the altitude from B to side AC is hb. Aryabhata on the grounds of Inter- Corollary 1. Each of the three altitudes in a University Centre for Astronomy and triangle gives a formula for the area of the Astrophysics triangle. Thus, we have three formulas for in Pune, the area of any triangle: Maharashtra, India Area of any triangle equals: He is known for = a x h = b x h = c x h a b c explaining the

lunar eclipse, and solar eclipse, rotation of earth on its axis, Problem 6. Use any of your previous reflection of light by moon, sinusoidal triangles. Draw its three altitudes and functions, solution of single variable calculate the area of the triangle in three quadratic equation, value of π correct to 4 different ways. Did you get exactly the same answer or approximately the same decimal places, of the answer? How do you explain this? Are the Earth to 99.8% accuracy. three formulas for the area wrong? Practical Facts. Although simple, our

Explanation. Most likely you got 3 close formula for the triangle’s area is only useful but not exactly the same answers. The if the height can be readily found, which is difference in the answers is explained by the not always the case. For example, the

surveyor of a triangular field might find it errors of drawing the altitudes and measuring the of the sides and the relatively easy to measure the length of each altitudes. You got approximations, but not side, but relatively difficult to construct a necessarily the exact area in all three cases. ‘height.’ Other methods are used in practice, The formulas hold and they are correct! depending on what is known about the . 3 | P a g e

©Zvezdelina Stankova, Berkeley Math Circle Director, July 23, 2014

RECAP 1: Grouping Words RECAP 2: New Vocabulary and Ideas Split the 49 words into groups and make 4 Check ALL correct answers. Explain your true statements discussed in this lesson. choice and provide details.

Use all given clues. 1. The foot of an altitude in a triangle:

I. Is where a perpendicular and a side of adjacent angle. a any any area area area the triangle intersect. base by II. Is the foot of a hill. calculated III. Is the intersection of the of a half height. hypotenuse is is is is in a rectangle. legs. IV. Is the intersection of the altitudes in a of of of of of of opposite triangle. product product V. Is a point on the side of a triangle. rectangle right right sides. VI. Looks like the heel of your foot when taking times triangle triangle two two you stand up. The The The The the the the the the the 2. The area of a right triangle:

I. Cannot be calculated unless you know 1. ______the hypotenuse of the triangle. II. Is half of the area of a suitable ______rectangle. ______III. Was first calculated by Aryabhata. IV. Is the product of two adjacent sides of 2. ______the triangle, divided by 2.

______V. Has only two distinct formulas that use the altitudes and sides of the triangle. ______VI. More than 50% of the above are true.

3. ______3. In standard notation:

______I. The feet of the altitudes are denoted by ha, hb, and hc. ______II. The vertices of the triangle are denoted by lower-case letters. 4. ______III. The sides of the triangle are denoted ______by upper-case letters. IV. The altitudes have a special notation ______that uses the side which is perpendicular to the altitude.

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