Question

how
am/ab = ap/ac = mp/bc

Answer 1

Look closely and we can find two similar triangles.

Triangle ABC and AMP is similar.

Since we are given ratios of sides :)

AM:AB=AP:AC

In a fraction, this is AM/AB=AP/AC

As a same way

AM:AB=MP:BC

This is AM/AB=MP/BC

I hope my words are clear :)

Answer 2

Solution

To Prove :

AM/AB = AB/AC = MP / BC

Firstly,

In ∆ABC and ∆AMP,

• A = A [Common Angle ]

• ABC = AMP = 90°

• AM / MB = AP/ CP [MP || BC ]

From ASA similarity,

∆ABC ≈ ∆AMP

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Thales' Theorem

• If two lines are parallel,then the ratio of the corresponding sides they divide is equal

• It is also known as Basic Proportionality Theorem

Consider AM/MB = AP/CP

» MB/AM + CP/AP

Adding 1 on both sides,

» MB / AM + 1 = CP/AP + 1

» (MB + AM)/AM = (CP + AP)/AP

» AB/AM = AC/AP

» AM/AB = AP/AC __________(1)

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Also,

Since ABC and AMP are similar triangles

• The ratio of their area would be equal to square of the ratio of the corresponding sides

For instance,

Ar(AMP)/Ar(ABC) = (MP/BC)² = (AM/AB)²

» (MP/BC)² = (AM/AB)²

» MP/BC = AM/AB________(2)

From relations (1) and (2),

AM/AB = AP/AC = MP/BC

Hence,ProveD

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