Quadrilaterals. Listen, geometry is not necessarily a favorite topic in mathematics for a lot of people. So, now you're thinking, "What about you, Lori? How do you feel about geometry?" Well, it's not my favorite either 😅; I'm not in love with geometry. 🤷🏿 I will say, I am very visual, so I love the fact images, and the need to draw pictures, are a crucial part of geometry. I have lots of respect for this topic. Geometry is everywhere.
This week's problem was taken from the last administered NYS Regents High School Examination in Geometry before quarantine--January 2020, Part III, question 32. To solve this week's problem in completion, you need to recall the following math skills:
✔️ Plotting points on the coordinate plane
✔️ Slope, distance, midpoint formulas
✔️ Properties and theorems for quadrilaterals
Happy solving!
Check back on Friday, August 14th for the solution, which will be posted below ⬇️.
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✏️📓 Solution Time! 📓✏️
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In high school I learned the two-column proof method for geometry proofs. The proofs I remember doing the most were triangle proofs. I rarely, if ever, used the paragraph-style for writing geometric proofs. So, for this problem I'll complete the necessary computations and accompany them with statements that support those computations.
There are various way to prove that a quadrilateral is a rhombus. This is because there are various ways to prove that a quadrilateral is a parallelogram, of which a rhombus is a special case. I will prove that quadrilateral NATS is a rhombus with the following method:
I. Prove that quadrilateral NATS is a parallelogram
II. Prove that the parallelogram is a rhombus.
But first, here is an image of quadrilateral NATS (the rhombus in question):
**This plot was created using GeoGebra's Geometry App.
I. Prove that quadrilateral NATS is a parallelogram. I will do this by showing that the diagonals of NATS bisect each other. The midpoint formula was used.
Diagonal NT and diagonal AS share a midpoint. NM = TM and AM = SM. Therefore, diagonals NT and AS bisect each other. Therefore, quadrilateral NATS is a parallelogram. ✔️
II. Prove that parallelogram NATS is a rhombus. I will do this by showing that the diagonals are perpendicular.
The slopes of diagonal NT and diagonal AS are negative reciprocals of one another. Therefore, diagonal NT and diagonal AS are perpendicular. Therefore, parallelogram NATS is a rhombus. ∎
It's been quite a while since I've done one of these. This has me thinking 🤔 back to when I took my high school math Regents. I remember doing logic proofs. Maybe I'll explore that topic in a future WMP!
◾️ How do you feel about geometry??
◾️ What method did you use to complete the proof??
◾️ Comment below with your responses and let me know what you thought about this week's problem.
Without further ado, on to WMP! #16
Cheers!
The Younge Lady
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