View the step-by-step solution to:

If the midpoints of the sides of a parallelogram taken in succession are joined, the quadrilateral formed is a parallelogram. Synthetic proof and...

If the midpoints of the sides of a parallelogram taken in succession are joined, the quadrilateral formed is a parallelogram. Synthetic proof and prove.

Introduction: Synthetic and analytic techniques can be used to prove and derive many geometric relationships. Additionally, a straight edge and compass can be used to construct a variety of well-known geometric relationships. Task: Theorem : If the consecutive midpoints of the sides of a parallelogram are joined in order, then the quadrilateral formed from the midpoints is a parallelogram. B. Prove the theorem given above in Euclidean geometry using synthetic techniques. 1. Include each step of your proof. 2. Provide written justification for each step of your proof. My answer below. They approved the math as appropriate but kicked back the proof twice. I need improved proofs and revised math as necessary: To start the synthetic process I have drawn a line between point A and point C and another line between point B and point D. ▲ABC One of the two triangles created by adding the line between A and C. BE = EA The midpoint E is equally located between A and B. So, the line segments BE and EA are the same length. BF = FC The midpoint F is equally located between B and C. So, the line segments BF and FC are the same length. ?? ?? = ?? ?? The line segment EF. ±² || °³ Using the triangle proportion property, it shows that EF is parallel to AC.
Background image of page 1
▲ACD The second triangle from adding the AC line. DH = HA The midpoint H is equally located between A and D. So, the line segments DH and HA are the same length. DG = GC The midpoint G is equally located between C and D. ?? ?? = ?? ?? The line segment HG. So, the line segments DG and GC are the same length. HG || AC By bisecting two side of a triangle, the bisecting line is parallel to the third side; it shows that HG is parallel to AC. HG||AC and EF||AC so, HG||EF Proof that the HG side is parallel to the EF side. ABD One of the two triangles created by adding the line between B and D. BE = EA The midpoint E is equally located between A and B. So, the line segments BE and EA are the same length. AH = HD The midpoint H is equally located between A and D. So, the line segments AH and HD are the same length. ?? ?? = ?? ?? The line segment EH. E H G F A B C D
Background image of page 2
Show entire document
Sign up to view the entire interaction

Top Answer

Friend! Here is the synthetic proof... View the full answer

parallelogram 1.docx

Suppose ABCD is a quadrilateral and G, F, E, H are the midpoints of the sides AB, BC, CD,
and DA respectively.
If ABCD has the opposite sites parallel, then it is nothing but a rectangle and thus,...

Sign up to view the full answer

Why Join Course Hero?

Course Hero has all the homework and study help you need to succeed! We’ve got course-specific notes, study guides, and practice tests along with expert tutors.

-

Educational Resources
  • -

    Study Documents

    Find the best study resources around, tagged to your specific courses. Share your own to gain free Course Hero access.

    Browse Documents
  • -

    Question & Answers

    Get one-on-one homework help from our expert tutors—available online 24/7. Ask your own questions or browse existing Q&A threads. Satisfaction guaranteed!

    Ask a Question
Ask a homework question - tutors are online