Question

On your own paper, write or type out the following proofs. Be sure to include all statements and reasons.Prove the Triangle Proportionality Theorem using a flow proof, paragraph or two-column proof.

Answer 1

Consider the triangle ABC.

If we draw a line PQ parallel to BC according to the triangle proportionality theorem the ratio of AP to PB is equal to the ratio of AQ to QC

We are using the two-column proof method.

Join the vertex B and Q and also Join the vertex P and C.

Draw a perpendicular line from Q to AB , and mark the point as N.

Draw a perpendicular line from P to AC , and mark the point as M.

`1\ldots..Area\text{ of }\Delta APQ=(1)/(2)* AP* QN\ldots\ldots\ldots\ldots\text{.}area\text{ of triangle=}(1)/(2)* base* height.`

QN is the height of the triangle PBQ and APQ.

`2\ldots..Area\text{ of }\Delta PBQ=(1)/(2)* AB* QN\ldots\ldots\ldots\ldots\text{.}area\text{ of triangle=}(1)/(2)* base* height.`

PM also the height of the triangle APQ.

`3\ldots..Area\text{ of }\Delta APQ=(1)/(2)* AQ* PM\ldots\ldots\ldots\ldots\text{.}area\text{ of triangle=}(1)/(2)* base* height.`

`4\ldots..Area\text{ of }\Delta QCP=(1)/(2)* QC* PM\ldots\ldots\ldots\ldots\text{.}area\text{ of triangle=}(1)/(2)* base* height.`

`5\ldots..\frac{Area\text{ of }\Delta APQ}{Area\text{ of }\Delta PBQ}=((1)/(2)* AP* QN)/((1)/(2)* AB* QN)=(AP)/(AB)\ldots\ldots\ldots\ldots\text{.from (1) and (2)}`

`6\ldots..\frac{Area\text{ of }\Delta APQ}{Area\text{ of }\Delta\text{QCP}}=((1)/(2)* AQ* PM)/((1)/(2)* QC* PM)=(AQ)/(QC)\ldots\ldots\ldots\ldots\text{.from (3) and (4)}`

`8\ldots..(AP)/(PB)=(AQ)/(QC)\ldots\ldots\ldots\ldots\text{.from (5), (6) and (7)}`

Hence proved the theorem.