proof of midpoint theorem
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Step-by-step explanation:
this theorem hold? Yes, it will, and the proof of the converse is presented next.
Converse of the Midpoint Theorem
The Converse of the Midpoint Theorem says that the line drawn through the midpoint of one side of a triangle which is parallel to another side will bisect the third side. Consider a triangle ABC, and let D be the midpoint of AB. A line through D parallel to BC meets AC at E, as shown below:
Converse of the Midpoint Theorem
The Converse of the Midpoint Theorem says that E is the midpoint of AC.
Proof of the Converse of the Midpoint Theorem
Proof: Suppose that E is not the midpoint of AC. Let F be the midpoint of AC. Join D to F, as shown below:
Midpoint Theorem Exercise
By the Midpoint Theorem, DF || BC. But we also have DE || BC. This cannot happen because through a given point (in this case, D), exactly one parallel can be drawn to a given line (in this case, BC).
Thus, E must be the midpoint of AC. This completes our proof.
Joining the midpoints the sides of a triangle
An interesting consequence of the midpoint theorem is that if we join the midpoints of the three sides of any triangle, we will get four (smaller) congruent triangles, as shown in the figure below:
Four smaller congruent triangles
We have:
ΔADE
Δ
A
D
E
≡ ΔFED
Δ
F
E
D
≡ ΔBDF
Δ
B
D
F
≡ ΔEFC
Proof: Consider the quadrilateral DEFB. By the midpoint theorem, we have:
1. DE = ½ BC = BF
2. DE || BF
Thus, DEFB is a parallelogram, which means that ΔFED ≡ ΔBDF.Similarly, we can show that AEFD and DECF are parallelograms, and hence all the four triangles so formed are congruent to each other (make sure that when you write the congruence relation between these triangles, you get the order of the vertices correct).
Solved examples on the Midpoint Theorem
Example 1. Consider a triangle ABC, and let D be any point on BC. Let X and Y be the midpoints of AB and AC.
Mid point theorem example
Show that XY will bisect AD.
Solution. By the midpoint theorem, XY || BC. Now, consider ΔABD. The segment XE is parallel to the base BD, and X is the midpoint of AB. By the converse of the midpoint theorem, E must be the midpoint of AD.
Thus, XY bisects AD.
Example 2. Prove that if three parallel lines make equal intercepts on one transversal, then they will make equal intercepts on any other transversal as well.
Solution. Let us first understand the problem better. Consider three lines and two transversals, as shown below:
Mid point theorem example
Suppose that the intercepts on the left transversal are equal, that is, AB = BC. We then have to prove that the intercepts on the right transversal will also be equal, that is, DE = EF.
To prove this, join A to F:
Intercepts on transversal are equal
Consider ΔACF. Since B is the midpoint of AC and BG || CF, the (converse of the) midpoint theorem tells us that G is the midpoint of AF.
Now, consider ΔAFD. We have shown that G is the midpoint of AF. Also, GE || AD.Thus, the (converse of the) midpoint theorem tells us that E must be the midpoint of FD. In other words, DE = EF.
Example 3. Consider a parallelogram ABCD. E and F are the midpoints of AB and CD respectively. Show that the line segments AF and EC trisect the diagonal BD.
Solution. Consider the following figure:
Parallelogram - mid point theorem
We have to show that
BX = XY = YD = BD/3
First of all, we note that AECF is a parallelogram (why?), and thus, EC || AF. Now, consider ΔBAY:
Parallelogram - mid point theorem
Since E is the midpoint of AB, and EX || AY, the (converse of the) midpoint theorem tells us that X is the midpoint of BY, which means that BX = XY. Similarly, we can prove that XY = YD. Thus,
BX = XY = YD = BD/3
Step-by-step explanation:
Mid Point Theorem
Geometry is one among the fundamental and essential branches of mathematics. This field deals with the geometrical problems and figures which are based on the properties. One of the important theorems in the field of geometry that deals with the properties of triangles are called the Mid- Point Theorem.
The theory of midpoint theorem is used in the coordinate geometry stating that the midpoint of the line segment is an average of the endpoints. To solve an equation using this theorem, it is important that both the ‘x’ and the ‘y’ coordinates should be known. The Mid- Point Theorem is also useful in the fields of calculus and algebra.
The midpoint theorem states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”
MidPoint Theorem Proof
If midpoints of any of the sides of a triangle are adjoined by the line segment, then the line segment is said to be in parallel to all the remaining sides and also will measure about half of the remaining sides.
Consider the triangle ABC, as shown in the above figure,
Let E and D be the midpoints of the sides AC and AB. Then the line DE is said to be parallel to the side BC, whereas the side DE is half of the side BC; i.e.
DE∥BC
DE = (1/2 * BC).
Now consider the below figure,
Construction- Extend the line segment DE and produce it to F such that, EF=DE.
In the triangle, ADE, and also the triangle CFE
EC= AE —– (given)
∠CEF = ∠AED {vertically opposite angles}
EF = DE { by construction}
hence,
△ CFE ≅ △ ADE {by SAS}
Therefore,
∠CFE = ∠ADE {by c.p.c.t.}
∠FCE= ∠DAE {by c.p.c.t.}
and CF = AD {by c.p.c.t.}
The angles, ∠CFE and ∠ADE are the alternate interior angles. Assume CF and AB as two lines which are intersected by the transversal DF.
In a similar way, ∠FCE and ∠DAE are the alternate interior angles. Assume CF and AB are the two lines which are intersected by the transversal AC.
Therefore, CF ∥ AB
So, CF ∥ BD
and CF = BD {since BD = AD, it is proved that CF = AD}
Thus, BDFC forms a parallelogram.
By the use of properties of a parallelogram, we can write
BC ∥ DF
and BC = DF
BC ∥ DE
and DE = (1/2 * BC).
Hence, the midpoint theorem is Proved.
MidPoint Theorem Formula
In Coordinate Geometry, midpoint theorem refers to the midpoint of the line segment. It defines the coordinate points of the midpoint of the line segment can be found by taking the average of the coordinates of the given endpoints. The midpoint formula is used to determine the midpoint between the two given points.
If P1(x1, y1) and p2(x2, y2) are the coordinates two given endpoints, then the midpoint formula is given as:
Midpoint = ((x1+x2)/2, (y1+y2)/2)
The converse of MidPoint Theorem
The converse of the midpoint theorem states that ” if a line is drawn through the midpoint of one side of a triangle, and parallel to the other side, it bisects the third side”.
Midpoint Theorem Example
The example is given below to understand the midpoint theorem.
Example:
In a triangle ABC, the midpoints of BC, CA, AB are D, E, and F respectively. Find the value of EF, if the value of BC = 14 cm
Solution:
Given: BC = 14 cm
To find the value of EF.
If F is the midpoint of AB and E is the midpoint of AC, then we can write it as:
EF = 1/2 (BC)
Now, substitute the value of BC
EF = 1/2(14)
EF = 7 cm
Therefore, the value of EF = 7cm.
The Mid- Point Theorem can also be proved using triangles. The line segment which is on the angle, suppose two lines are drawn in parallel to the x and the y-axis which begin at endpoints and also the midpoint, then the result is said to be two similar triangles. This relation of these triangles forms the Mid- Point Theorem.