Grade 7/8 Math Circles November 15 & 16, Areas of Triangles

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1 Faculty of Mathematics Waterloo, Ontario NL 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles November 15 & 16, 016 Areas of Triangles In today s lesson, we are going to learn how to find the area of a triangle in a variety of different ways. Before we get into that, let s review what we know about triangles so far. Triangles: A Review A triangle, denoted as ABC, has the following properties: ABC has 3 vertices and 3 sides, and 3 angles The sum of the interior angles of ABC is 180. We can label a triangle, ABC, as follows: A c b B y a z C Sum of interior angles = 180 = BAC + ABC + BCA = + y + z 1

2 3 Types of Triangles Equilateral Isosceles Scalene All side lengths are equal Two side lengths are equal All side lengths are different All angles are equal Two angles are equal All angles are different Find the Area of a Triangle Method 1: Basic Formula h A Basic Formula for Area of a Triangle Let b be the base of the triangle and h be the height of the triangle. The area of the triangle, A ABC, is... B b C A ABC = b h = 1 (b h) Eample Find the area of DEF. D 13 m E 5 m F By Pythagorean s Theorem, 13 = 5 + = 1 m So, A DEF = 5 1 = 30m

3 Method : Heron s Formula A Heron s Formula c Let a, b, and c be the side lengths of ABC. Then, b A ABC = s(s a)(s b)(s c) B a C where s = 1 (a + b + c) is the semi-perimeter of ABC. Eample 1 Find the area of P QR: s = 1 ( ) = 4 P 10 cm A P QR = 4(4 10)(4 1)(4 17) = 4(14)(3)(7) = 7056 A P QR = 84 cm 17 cm 1 cm R Q Eample Suppose you have ABC. What is the area of XY Z? (Image is not to scale.) C 10 X A 11 Z 16 1 Y 3 B Notice that AXY is isosceles XY = 16. Then, we see that AX = XC = ZX = Y Z = 10. Now we can use Heron s formula as follows: s = 1 (XY + Y Z + ZX) = 1 ( ) = 18 A XY Z = 18(18 16)(18 10)(18 10) = 18()(8)(8) = 304 = 48 units The area of XY Z is 48 units. 3

4 Eercise Challenge Three identical isosceles triangles are arranged as shown, with points A, B, C, and D lying in the same straight line. If AR = AD, what is the area of ADR? P Q R A 5 B C D Since AP B, BQC, and CRD are all identical isosceles triangles, we have that 5 = AB = BC = CD and P A = RD. Then, we can determine that AD = AB + BC + CD = = 15. We also know that AR = AD, so AR = 15. Now we can use Heron s formula to calculate the area of ADR: s = 1 ( ) = 4 A ADR = 4(4 15)(4 15)(4 18) = 4(9)(9)(6) = A ADR = 108 units The area of ADR is 108 units. 4

5 Method 3: Complete the Rectangle Suppose we have the coordinates of ABC. We do not know what the side lengths are but we are given the coordinates, kind of like the game Battleship! Draw the triangle on the graph below. A(,3) B(0,0) C(4,1) 3 A 1 Row 3 1 B Column 3 4 C 1 Let s find the area of ABC. Since we do not know the side lengths, we will enclose ABC inside a rectangle. Notice the rectangle is made up of four triangles: ABC, 1,, and 3. How can we calculate the area of ABC? Subtract the areas of 1,, and 3 from the area of the rectangle! A rectangle = length width = 4 3 = 1 A 1 = b h A = A 3 = 4 1 = 3 = = = 3 A ABC = A rectangle A 1 A A 3 A ABC = 1 3 A ABC = 5 units 5

6 Complete the Rectangle If we have a triangle, ABC, and we enclose it in a rectangle, then A ABC = A rectangle A 1 A A 3 where 1,, and 3 are triangles that form a rectangle with ABC. Eample Mr. Bean has a 7 m 9 m backyard and decides to construct a triangular-shaped patio. He plots his backyard plan on a graph as follows: 7 m 6 4 m 4 m 5 Row m 4 m m m 1 m Column What is the area of his patio? How much space does he have left in his backyard? A patio = A rectangle A 1 A A 3 = (5 8) 5 4 = = 18 m The area of Mr. Bean s patio is 18 m. After building his patio, he will have (7 9) 18 = = 45 m of his backyard left. 6

7 Method 4: Shoelace Theorem Also known as Shoelace Formula, or Gauss Area Formula Shoelace Theorem (for a Triangle) Suppose a triangle has the following coordinates: (a 1, b 1 ), (a, b ), (a 3, b 3 ) where a 1, a, a 3, b 1, b, and b 3 can be any positive number. Then, a 1 b 1 A 3 = 1 a b a 3 b = 1 3 (a b 1 + a 3 b + a 1 b 3 ) (a 1 b + a b 3 + a 3 b 1 ) a 1 b 1 where a is called the absolute value of a. (i.e. 3 = 3, 4 = 4) Eample Given the coordinates below, what is the area of ABC? A(3,4) B(7,9) C(11,4) A 3 = = 1 ( ) ( ) = 1 ( ) ( ) = = 1 40 = 1 40 A 3 = 0 m The area of ABC is 0 m. 7

8 We can actually generalize the Shoelace Theorem and use it to find the area of a shape with any number of sides! Shoelace Theorem (for a shape with n sides) Suppose a n sides, so there are n sets of coordinates: (a 1, b 1 ), (a, b ),..., (a n, b n ). Then, a 1 b 1 A n = 1 a b.. = 1 (a b 1 + a 3 b a 1 b n ) (a 1 b + a b a n b 1 ) a n b n a 1 b 1 Eample Calculate the area of a heagon with the following coordinates: A 6 = (3,9), (9,9), (11,5), (9,1), (3,1), (1,5) = 1 ( ) ( ) = 1 ( ) ( ) = = 1 18 = 1 18 A 6 = 64 units The area of the heagon is 64 units. 8

9 Angles, Angles & More Angles Acute Obtuse Right < 90 > 90 = 90 Complementary vs. Supplementary Angles Complementary Supplementary y y 90 = + y 180 = + y Opposite Angles are the angles opposite of each other when two lines intersect. z w y For the eample on the left, (w, y) and (, z) are our pairs of opposite angles. This means that... w = y = z Corresponding Angles occur when there is a line intersecting a pair of parallel lines. Z pattern F pattern C pattern > > > > y > y > y = y = y 180 = + y 9

10 Eample 1 Find all the missing angles,, y, and z, of the following image: (Note: Image is not to scale. Do not use a protractor.) > Solutions may vary. Using supplementary angles, we know 180 = 60 + z + 50 z = 70 > Then, by the Z pattern, we see that y = 50 Finally, by the sum of interior angles, we also know that = = 60 So, = 60 y = 50 z = 70 Eample In the diagram, P QR and ST U overlap so that RTQU forms a straight line segment. What is the value of? S P 50 R 30 T Q U The supplementary angle where PR intersects SU is = 130. And since the sum of interior angles is 180, we have that 180 = = = 0 Therefore, = 0. 10

11 Problem Set Solutions 1. Find the area of the following triangles: (a) A 3 cm (b) B cm C A ABC = 3 = 3 cm E 4 mm D F Use Heron s Formula: s = 1 ( ) = 6 A DEF = 6()()() = mm (c) DEF where D(, 8), E(8, 0), F (4, 4) 8 A DEF = = 1 ( ) ( ) 8 = = 1 8 A DEF = 4 units 11

12 . Find all the missing angles of the following diagrams and eplain your reasoning: Reasoning for solutions may vary. (a) > > Since a is a supplementary angle of 10, a = 60. By the Z pattern, b = 45. Using the sum of interior angles, c = (b) By the sum of interior angles, the missing angle of the triangle is = 50. Using opposite angles, = 50. (c) > > 50 By the C pattern, p = = 65. By the Z pattern, q = 50. 1

13 3. Given the perimeter, P ABC = 4 cm, what is the area of ABC? B 15 cm 14 cm A 13 cm C First, we need to find using the perimeter. P ABC = 4 = = 14 cm Now, we can use Heron s formula to find the area. s = 1 ( ) = 1 cm A ABC = 1(1 14)(1 15)(1 13) = 1(7)(6)(8) = 7056 = 84 cm 4. Given DE = 1, EF = 16, and F D = 0, what is the area of DEF? Using Heron s Formula, we get: s = 1 ( ) = 4 A DEF = 4(4 1)(4 16)(4 0) = 4(1)(8)(4) = 916 = 96 units 13

14 5. The perimeter of ST U is 3 cm. If ST U = SUV and T U = 1 cm, what is the area of ST U? Drawing ST U gives us: S T 1 cm U Since ST U = SUV, ST U is an isosceles triangle which means that ST = US. Use the perimeter to solve for ST and US. P ST U = 3 = 1 + ST + US = 1 + ST ST = Using Heron s Formula, we can find the area of ST U: 3 1 = 10 cm s = 1 ( ) = 16 A ST U = 16(16 1)(16 10)(16 10) = 16(4)(6)(6) = 304 = 48 cm 6. Find the area of a rectangle with the following coordinates: (,3), (6,3), (6,8), (,8) Using Shoelace Theorem, we get that... 3 A 4 = = 1 ( ) ( ) = = 1 40 = 1 (40) A 4 = 0 units 14

15 7. Peter Pan and the Lost Boys go looking for treasure in Neverland. They stole the map from Captain Hook but they can read is that the hidden treasure is located somewhere between Mermaid Lagoon (4,10), Pirates Cove (18,7), and Crocodile Creek (0,14). How much area, in kilometres, does Peter Pan and the Lost Boys have to search? Use the Shoelace Theorem! A 3 = = 1 ( ) ( ) = = 1 36 = 1 (36) A 3 = 18 km Peter Pan and the Lost Boys has 18 km of area to search. 15

16 8. * Bernadine and Fred are playing Battleship: Geometry Edition where ships sink with one hit. Fred has a triangle-shaped ship with the following coordinates: A(1,7), B(4,7), C(1,) and Bernadine has a pentagon-shaped ship with the following coordinates: P (4,1), Q(3,3), R(5,y), S(7,3), T (6,1) A Fred s Battle Grid B Bernadine s Battle Grid R C Q S P T (a) Fred s ship is in the shape of a right-angled triangle. If the area of Fred s ship is 6 units, what is the value of? Draw the rest of Fred s ship on his grid. Since we know the area is 6 units and the triangle is a right-angled triangle, we can use the basic formula for area to find the height, h(= BC). We can say AB (= 3) units is the base of our triangle (i.e. b = AB). A ABC = 6 = 3 h 6 = 3 h / / 1 3 = /3 h /3 4 units = h Since the height of Fred s triangle-shaped ship is 4 units, = 7 4 = 3. Thus, C(1, ) = C(1,3). (b) Bernadine s ship is in the shape of a pentagon. If the area of Bernadine s ship is 8 units, what is the value of y? Draw the rest of Bernadine s ship on her grid. 16

17 Since we know the area of the pentagon-shaped ship is 8 units and we know all coordinates ecept for R, we can use the Shoelace Theorem and solve for y A P QRST = 1 5 y Therefore, y = 4. 8 = 1 ( y ) ( y ) 8 = 1 (7y + /40) (3y + /40) 8 = 1 4y 8 = 4y / / 16 4 = /4y /4 4 = y (c) If Fred attacked Bernadine at (4,3), does Fred sink Bernadine s ship? No, he does not. Unfortunately, Fred has poor aim. (d) If Bernadine attacked Fred at (,6), does Bernadine sink Fred s ship? Yes, Fred s ship sinks. 17

18 9. * In the diagram, what is the area of quadrilateral ABCD? A D 5 B 9 C Draw line segment AC to create two right-angled triangles, ABC and ACD. By Pythagorean s Theorem, AC = = Again, by Pythagorean s Theorem, AD = AC 5 = 50 5 = 5 = 15. The area of ABCD is the sum of the areas of ABC and ACD. Both triangles are right-angled, so we can use the basic formula to find the area of both triangles. A ABCD =A ABC + A ACD = = = 19 = 96 units 18

19 10. ** In the diagram, P QR is isosceles with P Q = P R = 39 and SQR is equilateral with side length 30. What is the area of P QS? (Round your answer to the nearest whole number.) To find the area of P QS, we will calculate the area of P QSR by subtracting the area of SQR from P QR. Use Heron s formula to calculate the area of both triangles. s P QR = 1 ( ) = 54 A P QR = 54(54 39)(54 39)(54 30) = 54(15)(15)(4) = = 540 s SQR = 1 ( ) = 45 A SQR = 45(45 30)(45 30)(45 30) = 45(15)(15)(15) = Now, A P QSR =A P QR A SQR = = The area of P QS is half of the area of P QSR since P Q = P R, SQ = SR, and P QS shares side length P S with P RS. A P QS = 1 A P QSR = = 75 Therefore, the area of P QS is 75 units. 19

11. *** A star is made by overlapping two identical equilateral triangles, as shown below. The entire star has an area of 36. What is the area of the shaded region?

Each smaller triangle has two equal side lengths, marked by the single tick, so all 6 of the smaller triangles are identical isosceles triangles.

20 11. *** A star is made by overlapping two identical equilateral triangles, as shown below. The entire star has an area of 36. What is the area of the shaded region? Since the triangles are equilateral, each angle of both triangles is 60. Each smaller triangle has two equal side lengths, marked by the single tick, so all 6 of the smaller triangles are identical isosceles triangles. Since the smaller triangles are isosceles, the 1 missing angles can be calculated as follows: ( ) = 60. Because each angle of the small triangle is 60, all the small triangles are identical equilateral triangles as shown below. Now, all side lengths of the inner heagon are equal, so it is a regular heagon. This mean each angle is equal to 10. Since a regular heagon is symmetrical, drawing the 3 diagonals of the heagon creates 6 more small triangles as shown below. As it turns out, the small triangles in the inner heagon are also identical equilateral triangles. (The diagonals split the angles of the heagon in half, 10 = 60. The third angle in the small triangles also equal 60.) The star is made of 1 small identical equilateral triangles so the area of each small triangle is 36 1 = 3. The shaded region is made of 9 small triangles and so the area of the shaded region is 9 3 = 7. 0

Grade 6 Math Circles March 22-23, 2016 Contest Prep - Geometry

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles March 22-23, 2016 Contest Prep - Geometry Today we will be focusing on how to solve