Old friend Olivia visited Sara
Nana told Sara that Olivia phoned and wanted to come and visit her on Saturday. Olivia was with Sara in her grades six through eight but then Olivia’s parents moved to another district in the city, and the two friends ended up going to different schools. Even though they were good friends, somehow they didn’t connect after that even though Olivia had all the contact information for Sara. It had been almost two years since they had last seen each other. Nana also had fond memories of this girl.
Sara: Thanks Nana. Is this the Olivia Sanchez from my grade eight?
Nana: Yes, and I told her that she could come on Saturday morning. I would have samosas ready for her. She used to love them.
Sara: Yes, she loved them. Do you know that she called them Indian empanadas? I will make sure that I remain home on Saturday to be with Olivia for the whole day if she wishes.
Olivia came over as expected. She had grown a little bit as expected but nothing out of line. She still had a short figure, bounced a lot and was as chirpy as ever. The two friends were so glad to see each other. They reminisced as if they were together even yesterday. They had so many memories to share about every boy and girl in that middle school and they also had questions about where everyone was now, what they did and who they were dating. Of course, as teen age girls, they were both interested in talking about boys. Olivia had a few boys as friends but none who she thought was a boyfriend material. She was thrilled to hear that Sara had found Johnny.
Olivia: So how did you and Johnny meet?
Sara: He lives nearby on my way to school and rides his bike to the school every day. He saw me a few times on the way to school, got off his bike and started to walk with me. Then we met again in the cafeteria. One thing led to another and now we walk to school together all the time. I guess he liked what he saw.
Olivia: Is he cute?
Sara: Listen, I haven’t seen your new school. How far is it?
Olivia: A little less than three kilometers from here but you didn’t answer my question about Johnny.
Sara: I didn’t answer because it was a silly question. Tell you what. I want to see your new school. We will walk there and back. I will ask Johnny if he wants to come along. That way you can meet him and give me the answer to your own question.
Sara called Johnny. He was more than happy to come for a walk with them. They walked together to Olivia’s school. Of course, Johnny came with his new bike that had all the gizmos. As the school was about three kilometers away, the walk gave the two girls a little more time to gossip and to find out about each other.
Olivia’s school was triangular
When they reached Olivia’s school, Sara and Johnny were amazed at the shape of the building. Sara and Johnny’s high school had a circular building but this one had a triangular base. The front of the school ran East-West. The left side wall was at an angle, and the right side was so narrow that it almost had no wall.
Sara: How big is this school, like what is its area?
Olivia: I don’t know. I never thought of it.
Johnny was all excited. He had a new project for his bike – to find the area of the school and he told Sara and Olivia about it.
Olivia: You cannot measure the length of the back wall of the school. There is a fence that stops you from getting to the back. On the other side of the fence, there is a big company with high security locks on the gate of their compound.
Sara smiled as if she knew something
Sara smiled as if she knew something, and told Olivia not to worry. Johnny used his bike to measure the length of the front wall to be 100 meters. He also found that the angle between the front and side walls was 105°, and that the length of the side wall was 70 meters. He thought that was all he could measure because of the different restrictions but Sara told him that this was enough. They could not go inside the school because it was locked on account of it being a Saturday. So they decided to stroll back to Sara’s home. Sara and Olivia continued with their chitchat. After all, they had a lot to catch up after the two years. Johnny was not really into that conversation but then he had his bike to keep him engaged and happy.
The three of them came back to Sara’s home. Of course, Nana had the samosas ready for them. They were relishing the snacks when Olivia interrupted and asked as to how they were going to find the area of her school.
Johnny drew this picture (Fig. 5.1) and said: The school base is a triangle with the front wall of 100 meters, a side wall of 70 meters and the angle of 105° between the two walls. Sara, what do we do next? You said, this was enough information.
Sara tweaked Johnny’s picture
Sara: Let’s tweak your picture a little bit. First, let us label it. AB is the front wall of the school, AC is the side wall and BC is the back wall. Angle BAC is 105°. I am going to draw a hypothetical line to extend the front all the way to D, and draw a vertical line from C to cross it at D such that the angle ADC is 90° (Fig. 5.2).
Johnny interrupted as soon as Sara drew these dotted lines: Now, this is very simple. From this picture, we can find both – the length of the back wall and the area of the school.
Olivia: I think so. Angle DAC= 75° because it is complementary to the angle BAC which is 105°. Using the lengths of the sides:
CD/CA = sin 75° and DA/CA = cos 75°. What are the values for sin and cos for 75°.
Johnny: sin 75° = 0.9659 and cos 75° = 0.2588. That means CD = 0.9534 x CA = 0.9659 x 70 or 67.61 meters and DA = 0.2588 x CA = 0.2588 x 70 = 18.12 meters.
Olivia: That’s neat. We have two right angle triangles – One is DAC made on the left side of the school. It has a height of 67.61 meters and a base of 18.12 meters. The second is the bigger triangle DBC with the same height but a length of 100 + 18.12 =118.12 meters.
Dimensions of the school
Sara: So you remember the area of any triangle is height x base/2.
Johnny: Yes, I remember that from my geometry class. So the area of the school building is 67.61 x 100/2 (height x front wall/2) which is 3380.7 square meters. So that’s done. Olivia, I told you Sara will help us figure this out. Now what about the back wall.
Olivia: That’s easy. Remember the Pythagoras theorem from geometry.
Johnny: I remember it and also Sara reminded me of it two weeks ago. It is:
Hypotenuse2 = base2 + height2. For the triangle DBC, it would be
BC2 = DB2 + DC2 or BC2 = 118.122 + 66.74 2 or BC =136.1 meters.
So the back wall is 135.67 meters long.
Olivia: I am so happy and feel like jumping with joy. I am going to talk to the principal’s office and post the lengths of the walls on the school bulletin board. Hope you don’t mind. I will give you guys full credit for it.
Johnny: Don’t mention our names, please. Why would outside students come to your school for this type of measurements? Besides, all I did was to have some fun with my bike. Anyways, I have to go home because my mom must be waiting for me,
Sara: Olivia, you guys did all the calculations, you alone could have done them if we weren’t here. I only drew the dotted lines. Johnny is right. Don’t mention our names. However, there is one more thing that you should do before you talk to your school guys.
Olivia: What’s that?
Sara: What will be your answer if the Math teacher asks you why you gave only one angle and not all three of them?
Olivia: Because, we measured only one.
Determining all the angles
Sara showed Olivia the Trig book to show how she could get all the angles of this triangle. Olivia read it with wide eyes as if she didn’t believe that she could get the other two angles. She read out the rule called the law of sines: sin A/a = sin B/b = sin C/c and drew this picture. In this picture the angle A was 105° as Johnny had measured and the side a = 136.1 meters.
Because sin A = sin 105° = 0.9659, the value of sin A/a = .007097. Then the values of sin B/b and sin B/b must also be 0.007097. with c =70 meters, sin B = 0.007097 x 70 = 0.4968 or angle B = 29.8°. Sin C/c = 0.007097 = 0.7097 because c = 100 meters or angle C = 45.2°. Because 105 + 29.8+ 45.2 =180 which should be the sum of all angles of a triangle, the answer checked out.
Olivia: Thanks Sara. I will now post a note on the notice board with lengths of all three sides and angles for the school.
Sara: So all that’s fine. You didn’t tell me what you think of Johnny.
Olivia: Don’t make me jealous. He’s cute, smart and loves you. I wish I had a boyfriend like him.
Olivia saw the school principal and asked permission to put a note about what she had found out about school. The principal asked her to verify the information with her Math teacher. She did so and then wrote a note on these interesting facts about the school: obtuse triangular shape, lengths of walls, the angles between them and the area of the school building. Olivia got a good praise from her schoolmates. Some of them were curious about the details of how she determined all this. She told them everything. Of course, only her close friends got the nitty-gritty about her visit with Sara, and the details about Sara’s cute boyfriend.
Challenge: Since Johnny had done so well in school the previous semester, his parents got him a drone with a remote control. He wanted to fly it and asked Sara to come along to the park with him. Of course, Sara agreed because she loved being with him. She also brought her dad’s sextant along. Johnny was flying the drone and had it on a still spot and said to Sara who was sitting next to him, “I wonder how far the drone is.” Sara measured that the drone was at an angle of 45° from the ground from where she was. She walked about 20 meters towards the drone and now the angle from the ground was 60°. She would have had to walk even further to be underneath the drone. From this Sara told Johnny how far the drone was from him. Can you figure out what this whiz girl did?
Solution: Let’s draw a triangle from Sara’s first position at A, second position 20 meters away at B and the Drone at C (Fig. 10.5). Extend the line AB towards D.
Given: Distance AB = 20 meters, angle CAB = 45°, angle CBD = 60° (note that Sara would have to walk further towards the point D in order to be under the drone.
Now, in the triangle ABC, angle A = 45°, angle B = 180° – 60° = 120° (complimentary angle), and angle C = 180°- 45°-120° = 15° (sum of all angles of a triangle is 180°).
According to the trigonometric identity called the law of sines:
sin A/a = sin B/b = sin C/c.
Side opposing the angle B = b (AC), and opposing the angle C = c (AB = 20 meters).
From the law of sines, sin B/b = sin C/c or sin 120°/b = sin 15°/20 or 0.866/b = 0.259/20 or
b/0.866 = 20/0.259 or b = 0.866 x20/.259 = 66.9 meters.
Johnny’s drone is approximately 66.9 meters away from him.