 # Midpoint Theorem by using the Equal Intercepts Theorem – Statement, Proof and Examples

Know what is midpoint theorem and the purpose of using it. To find the missing values of the sides of the triangle, the midpoint theorem is used. Check the formulae, examples, and equal intercept theorem. Follow the method to solve the midpoint theorem and intercept theorem proof in the below sections. Find out all the concepts related to 9th Grade Math concepts on our website. Practice well and get a good score on the exam.

## Midpoint Theorem – Definition

The midpoint theorem defines that the midpoints joining any two sides of a triangle is parallel to the other side of the triangle and it is also congruent to half of the other side. Consider the below triangle △ABC. ABC is a triangle, and D and E are the midpoints of the sides AB and AC respectively. DE is the segment that connects the midpoints of two sides, where DE ∥∥ BC and DE segment is the half of the length of side BC.

Once we know the length of BC, then we can easily find the length of DE as it is half of BC. We can also find the length of sides DA, BD, EC and AE. As DE and BC are parallel, the distance between the line segments is equivalent.

### Midpoint Theorem Proof Statement: In the triangle, ▵ABC, AB, and AC are the sides of the triangle, and D and E are the midpoints of these sides respectively.

To Prove:
DE ∥∥ BC
DE = ½ BC

Proof:

Draw a parallel line CR to BA i.e., CR ∥∥ BA to meet DE which is produced at R.
∠EAD = ∠ECR (Alternate angles Pair) is the equation (1)
AE = EC (Since, E is the midpoint of side AC) is the equation (2)
∠AEP = ∠CQR (Vertically opposite angles) is the equation (3)
Thus, ▵ADE ⧤ ▵CRE (ASA Congruency rule)∠∠
DE = ½ DR is the equation (4)
But, AD = BD (Since, D is the midpoint of side AB)
Also, BD ∥∥ CR (by construction)
In quadrilateral BCRD, BD = CR and BD ∥∥ CR
Therefore, quadrilateral BCRD is a parallelogram.
BC ∥∥ DR or BC ∥∥ DE
Also, DR = BC (Since BCRD is a parallelogram)
½ DR = ½ BC

Hence, the midpoint theorem is proved.

### Equal Intercept Theorem Statement

The equal intercept theorem states that if there are two or three parallel lines and transversal makes equal intercepts on these lines, then any line cutting them will make equal intercepts. It defines that for any given three mutual parallel lines, the line that is passing through these lines forms intercepts with the distance between the lines of corresponding ratios. The equal intercept theorem is used in many geometrical proofs like converse midpoint theorem, midpoint theorem, etc.

#### Midpoint Theorem by using the Equal Intercepts Theorem Proof: Statement: Consider that l,m, and n are three perpendicular lines such that l ∥∥ m ∥∥ n. P is the transversal that cuts l, m, and n in A, B, and C respectively like AB = BC. q is also another transversal.

To Prove:
PQ = QR
Construction: Through Q, we have to draw a line r such that r ∥∥ p.

Proof:
With the above construction, pairs of opposite sides are parallel to each other and opposite sides of the parallelogram are equal.
ABQX is a parallelogram
As a pair of opposite sides are parallel
AB = XQ is the equation (1)
BCQY is also a parallelogram
As a pair of opposite sides are parallel
BC = QY is the equation (2)
From equations (1) and (2)
Vertically opposite angles From alternate angles, l ∥∥ n
BCQY is a parallelogram
AB = BCXQ = QY is the equation (3)
ASA Congruency CPCT (3)
Now, in △PQX and △RQY
∠PQX = ∠RQYX = QY
∠PQX = ∠RQYXQ = QY
∠PQX = ∠RYQ

The midpoint theorem and equal intercept theorem help to prove something about the triangle. The equal Intercept theorem is used in many geometrical proofs. Check the above article for the proofs and explanations of midpoint and equal intercept theorems.

### FAQs on Midpoint Theorem by using the Equal Intercepts Theorem

1. What does the midpoint theorem state?

The Midpoint theorem states that “ If a line segment of a triangle is joining the midpoints of 2 sides then that line segment is parallel to the third side and also half the length of the 3rd side.

2. What does midpoint mean?

A midpoint defines a point that lies in or at the middle or equal from both the ends of a line segment. In other words, the midpoint is halfway between the beginning and end.

3. What is the equal intercept theorem?

Equal intercept theorem defines that for any given three mutual parallel lines, the line that is passing through these lines forms intercepts with the distance between the lines of corresponding ratios.

4. Are intercepts made by parallel lines only?

No, any lines can make intercepts but only parallel lines can make equal intercepts.

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