Given: Angle ABC is a right triangle with hypotenus AB
D,E and F are midpts of the sides of the triangle
Proof: a. CDEF is a rectangle
b. EC is half of AB
To prove part a, we need to show that the opposite sides of the quadrilateral CDEF are parallel and equal in length.
1. Start by drawing a diagram of the triangle ABC with the midpoints D, E, and F labeled.
2. Recall that the midpoint of a side of a triangle divides that side into two equal segments. So, we know that AD = DB, BE = EC, and CF = FA.
3. Since AD = DB, we can conclude that segment AD is parallel to segment BC (by the Midpoint Theorem).
4. Similarly, using the Midpoint Theorem, we can conclude that segment BE is parallel to segment AC, and segment CF is parallel to segment AB.
5. Now, we have opposite sides of the quadrilateral CDEF (segments AD and CF) that are parallel to each other. Similarly, segments BE and DF are parallel to each other.
6. Moreover, since AD = DB and BE = EC, we can also conclude that the opposite sides of CDEF are equal in length. (Remember that D, E, and F are midpoints of the sides of the triangle.)
7. Therefore, by definition, quadrilateral CDEF is a rectangle. This completes the proof of part a.
To prove part b, we need to show that segment EC is half the length of the hypotenuse AB.
1. Recall that E is the midpoint of side AC. Therefore, segment AE is half the length of the hypotenuse AB.
2. Similarly, since E is the midpoint of side BC, segment BE is also half the length of the hypotenuse AB.
3. By the Transitive Property of Equality, we can conclude that segment AE is equal to segment BE.
4. Therefore, segment EC is half the length of segment AE, which is half the length of the hypotenuse AB.
5. Hence, we have proven that EC is half of the length of the hypotenuse AB.
This completes the proof of part b.
To prove the statements:
a. CDEF is a rectangle
b. EC is half of AB
We can use the properties of triangles and rectangles to justify each step of the proof.
Proof:
a. To show that CDEF is a rectangle, we need to prove that opposite sides are parallel and that all angles are right angles.
1. Since D, E, and F are the midpoints of the sides, we can use the Midpoint Theorem which states that a line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
2. Therefore, DF is parallel to AB and half its length, and EF is parallel to AC and half its length.
3. Since DF is parallel to AB and EF is parallel to AC, by the Converse of the Alternate Interior Angles Theorem, angles DFE and A are congruent, and angles EFD and C are congruent.
4. Since angle DFE is congruent to angle A, and angle EFD is congruent to angle C, we have two pairs of corresponding angles that are congruent, which demonstrates that DE is parallel to AB.
5. Similarly, using the corresponding angles, we can prove that CF is parallel to AB.
6. By proving that DF is parallel to AB and CF is parallel to AB, we can conclude that DF is parallel to CF.
7. Hence, all opposite sides of the quadrilateral CDEF are parallel, satisfying one condition for a rectangle.
8. To show that all angles in CDEF are right angles, we can analyze the diagonal AC.
9. AC is the hypotenuse of the right triangle ABC, and it is well-known that the diagonal of a rectangle divides it into two congruent right triangles.
10. Since triangle ABC is a right triangle with hypotenuse AB, angle A is a right angle.
11. AC is the diagonal of the rectangle CDEF, and therefore, angle C and angle E must also be right angles.
12. Since all angles in a rectangle are right angles, we have demonstrated that CDEF is a rectangle.
b. To show that EC is half of AB, we can use the properties of midpoints and perpendicular bisectors.
1. Since D and E are midpoints of AB and AC, respectively, we can use the Midpoint Theorem.
2. The Midpoint Theorem states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
3. Therefore, DE is parallel to BC and half its length, and EF is parallel to AC and half its length.
4. Since DE is parallel to BC, and EF is parallel to AC, by the Converse of the Alternate Interior Angles Theorem, angles C and DEF are congruent.
5. AB is the hypotenuse of right triangle ABC, and EF is parallel to AC, which is half the length of AB.
6. Therefore, EC is half of AB.
By proving that CDEF is a rectangle and EC is half of AB, we have completed the proof.