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Equal Intercepts Theorem
Intercept
In the figure given above, XY is a transversal cutting the line L1 and L2 at P and Q respectively. The line segment PQ is called the intercept made on the transversal XY by the lines L1 and L2.
If a transversal makes equal intercepts on three or more parallel lines then any other transversal cutting them will also make equal intercepts.
Given: Let there be three straight lines L1, L2, and L3 such that L1 ∥ L2 ∥ L3.
Transversal AB makes equal intercepts on L1, L2 and L3, I.e., PQ = QR. Another transversal CD makes intercepts KM and MN.
To Prove: KM = MN.
Construction: Join PN which cuts the L2 at O.
Proof:
|
Statement |
Reason |
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1. PQ = QR and QO ∥ line L3. |
1. Given. |
|
2. O is the midpoint of PN, i.e., PO = ON. |
2. By converse of Midpoint Theorem. |
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3. PO = ON and OM ∥ L1. |
3. By statement 2 and given. |
|
4. M is the midpoint of NK, i.e., KM = MN (Proved) |
4. By converse of Midpoint Theorem. |
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