Properties and Attributes of Triangles
A math story
(Based loosely on the novel Jane Eyre by Charlotte Brontë, except Edward Rochester isn't blind in my version and Jane is going to University to further advance her knowledge.)
Kayla Stephenson
5th Period
(Another note: Theorem names are in bold, definitions are underlined. Examples are normal, but paired with their theorems.)
"Ugh! I can't concentrate!" Jane muttered to herself, staring at the stacks of books in front of her. She was attempting to study for her Geometry exam, and she had an overwhelming amount of definitions to memorize.
Adèle looked up at her from where she was sitting on the floor, playing with a doll. "Je suis désolé. Would you like me to call Monsieur Rochester to help you?" the little girl asked in broken English.
Jane shrugged miserably and leaned her forehead on her open notebook, closing her eyes. She just needed a break, but the test was tomorrow. She was completely stressed, and was so lost in her thoughts that she didn't hear a new set of footsteps enter the room.
"Darling?" a deep voice asked. She jumped as she felt a big hand on her shoulder, and sat up immediately. Her husband, Edward, turned her so that she was facing him. "Whatever is the matter?"
Jane groaned. "My exam is tomorrow, and I cannot focus on the material." She looked wistfully out the window, and added, "Especially on such a beautiful day!" The trees were swaying slightly in the wind, and sunshine was pouring in through all of the elegant windows.
Edward furrowed his big eyebrows, thinking for a moment. "Well," he asked, "what are you studying?"
She glanced at her notes. "Properties and Attributes of Triangles," she replied.
His eyes lit up. "Well, I can certainly help you with that! I did rather well with Geometry in my day. Come, come, hand me your books, and grab a notebook and your pen kit." He took her books off of the desk and started out the door.
Jane grabbed her notebook, put a cap on the inkwell, and picked up her quill pen, then hurried after him. "Edward! Where are you going?" she called after him. He just laughed and headed out the front door.
She finally caught up with him at the bottom of the front steps, where he smiled and took her hand. "A desk isn't the only place to study, nay; I believe it to be the worst place to study. Follow me," he answered mysteriously.
They walked hand in hand through the large garden. It was a fine spring day, and all of the flowers were beginning to bloom. The sun was shining, and the breeze wasn't too cool. The birds were chirping loudly, and Jane could see a few building their nests. The pair finally came to an ivy covered wooden bench swing, and Edward took a seat, motioning for Jane to join him. She sat down, and he put his arm around her.
"Now," he said, "read over your notes, and then I will quiz you. Don't over-study or you'll stress yourself out. Read them once, and only once. Agreed?" He smiled at her, and she nodded in acquiescence.
After fifteen minutes, she looked up at him. "Ready? Don't tease me if I can't remember…" she frowned uncertainly.
Edward shook his head. "I would never dream of it, darling. Now, let's see…okay. Let's work with bisectors first. Explain to me the Perpendicular Bisector Theorem, and also the converse of it."
Jane pursed her lips, then replied, "The Perpendicular Bisector Theorem states that…if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. And the converse says that if a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment."
He smiled at her and said, "Very good, Jane. Now, how about the Angle Bisector Theorem and its converse?" he asked.
"If a point is on the bisector of an angle, then it is equidistant from the sides of the angle," she replied quickly. "And the converse is that if a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle."
"Wonderful!" he exclaimed. "Let's see…ah! What is the Circumcenter Theorem, and what part of the triangle is used to find the circumcenter?"
"Well, you use the perpendicular bisectors to find the circumcenter, correct?" She looked at Edward, and he nodded. Jane smiled, and then continued. "The Circumcenter Theorem states that…ah…the circumcenter of a triangle is equidistant from the vertices of the triangle."
"Excellent. Same question goes for the incenter."
Jane replied, with more confidence, "The angle bisectors are used to find the incenter of a triangle. The Incenter Theorem states that the incenter of a triangle is equidistant from the sides of the triangle."
"Correct. Once again, same question for the centroid of a triangle."
"The medians, which are made up of a midpoint and the opposite vertex, are used to find the centroid. And the Centroid Theorem states that the centroid is located…2/3 of the distance from each vertex to the midpoint of the opposite side."
"Wonderful job, darling!" Edward smiled and kissed her, then looked at her notebook again. "Alright, do you have your pen? Be uncapping the inkwell while you answer this. What is the Triangle Midsegment Theorem? Oh, do be careful, Jane. That dress is new."
Jane stuck her tongue out at him and began untwisting the cap as she thought. "A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side." She dipped the quill in the inkwell and sat it next to her.
"That's correct. Now, hand me the pen." He took the quill and began writing on a fresh page of the notebook. He sketched a triangle and put some values. "Solve this for me," he said, and handed it back to her.
"Well," Jane said, thinking out loud. "If the midsegment is half of the length of the side it is parallel to, then you would have to divide the side by 2." She wrote down the equation (1/2)3n=54, followed by 1.5n=54. She then divided 54 by 1.5, finally coming to the conclusion that n=36.
Edward smiled at her work. "Beautifully done, Jane. What is the Triangle Inequality Theorem?"
"The sum of any two sides' lengths of a triangle is greater than the third side length," she replied, raising an eyebrow as he handed her the notebook and pen once more.
"This is for your calculations. Could a shape be a triangle if its side lengths are 14, 18, and 32?"
Jane set to work, chewing on her lip. When she added 14 and 32, they were greater than 18. She put a checkmark next to that, and then moved on 18 and 32: they also received a checkmark. But 18 and 14 equaled 32, so she put a red x next to that equation, and said, "No sir, it cannot be a triangle."
Edward kissed her again and said, "Great work, Jane. Let's take a break from examples for a little bit. What is the Hinge Theorem and its converse?"
"The Hinge Theorem says that if two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle." She looked at Edward inquiringly, and he nodded in encouragement. "The Converse of the Hinge Theorem," she continued, "states that if two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side."
Edward nodded, saying, "Very good. Alright, we're almost done. What is the Pythagorean Theorem and its converse?"
Jane sighed at him sketching another triangle, but answered, "The Pythagorean Theorem is if a triangle is a right triangle, then the equation of its sides is a²+b²=c², a and b being the legs and c being the hypotenuse. The converse is if a²+b²=c², then a triangle is a right triangle."
Edward nodded, handing her a paper with this on it:
Jane took the paper, and wrote 4²+b²=8², then, simplifying it, wrote 16+b²=64. She then subtracted 16 from 64, (which equaled 48) and took the square root, getting 4√3 for her answer. Edward smiled, exclaiming, "Great job! Now, if a triangle has the sides 7, 10, and 12, is it an acute, obtuse, or right triangle?"
Jane added 7² and 10² and found the sum was 149. "Well, twelve squared is 144, which is less than 149, so it must be acute," she replied.
Edward smiled, and said, "Marvelous. Two more definitions, one more example, and you'll be done. What is the 45-45-90 Triangle Theorem?"
"In a 45-45-90 triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times the square root of two," Jane answered.
"And in a 30-60-90 triangle?"
"Then…the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times the square root of three. Right?"
"Right," Edward replied, drawing a 45-45-90 triangle. The length of the hypotenuse was fifteen, and he indicated that Jane was to find the length of the legs. She divided fifteen by the square root of two, and got 7.5√2 as her answer.
"Perfect! Jane, darling, you have absolutely nothing to worry about. You are beyond ready for exam." Edward stood up and held out his hand. Jane took it, smiling, and stood up, standing on her tiptoes to give him another kiss. "I'll have Mary make you some good luck meatloaf tonight. No more studying, alright?" Jane nodded, and he wrapped his arm around Jane's waist, pulling her to him as they walked back to their home.