197k views
3 votes
Am is a median in abc (m is in bc). a line drawn through m intersects ab at point p.

1 Answer

6 votes

Final Answer:

The areas are:

Area(∆APC) = 70 m²
Area(∆PMC) = 35 m²

Step-by-step explanation:

To find the areas of ∆APC and ∆PMC when given that Area(∆APM) = 35 m², let's consider the properties of a median and the fact that M is the midpoint of BC.

Since AM is a median, it divides triangle ABC into two triangles of equal area. Therefore, the area of ∆AMB is equal to the area of ∆AMC.

Now let's focus on line MP, which is given to intersect AB at its midpoint P. This implies that AP = PB. As a result, triangle APM is congruent to triangle BPM by the Side-Side-Side (SSS) congruency rule (since AP = PB, PM is a common side, and AM = MB by definition of a median).

Since ∆APM is congruent to ∆BPM, they have the same area. We are given that Area(∆APM) = 35 m², so Area(∆BPM) must also be 35 m².

Now, let's consider the larger triangles. Since M is the midpoint of BC, and P is the midpoint of AB, we have established that:

Area(∆AMB) = Area(∆AMC) (due to AM being a median)
Area(∆APM) = Area(∆BPM) (due to P being the midpoint of AB)

Adding the areas of ∆APM and ∆BPM gives us the area of the entire ∆ABM (because they are two non-overlapping triangles that together form ∆ABM):

Area(∆ABM) = Area(∆APM) + Area(∆BPM) = 35 m² + 35 m² = 70 m²

Since ∆AMB is equal in area to ∆AMC (because AM is a median), we also find that:

Area(∆AMC) = 70 m²

Now, to find Area(∆APC), we need to add the areas of ∆APM and ∆PMC. However, remember that ∆PMC is part of ∆AMC, which we already know has an area of 70 m².

Since ∆PMC and ∆APM both share side PM, we can express the area of ∆APC as the sum of the areas of ∆APM and ∆PMC:

Area(∆APC) = Area(∆APM) + Area(∆PMC)

We know Area(∆APM) is 35 m², and because ∆APM and ∆BPM are congruent and combine to form the same area as ∆AMC, Area(∆PMC) must also be 35 m².

Therefore:

Area(∆APC) = 35 m² + 35 m² = 70 m²

Complete question:

AM is a median in △ABC (M∈ BC ). A line drawn through point M intersects AB at its midpoint P. Find areas of △APC and △PMC, if Area of APM=35m².

Am is a median in abc (m is in bc). a line drawn through m intersects ab at point-example-1
User Opsidao
by
8.5k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories