Answer:

To prove that Q is the midpoint of AB and that PB = PA = 1/2 AC, we can use the properties of triangles and the fact that PQ bisects AB.

a) To prove that Q is the midpoint of AB, we can use the fact that PQ bisects AB. By definition, the midpoint of a line segment is the point that divides it into two equal parts. Since PQ bisects AB, it means that Q divides AB into two equal parts, so Q is indeed the midpoint of AB.

b) To prove that PB = PA = 1/2 AC, we can use the properties of similar triangles. Since ABC is a right-angled triangle at B, we have the following relationships:

1 ) AP/AC = BP/BC (By the definition of similar triangles)

2 ) AC = AP + PC (By the segment addition postulate)

Now, let's substitute the values of AP and PC from the first equation into the second equation:

AC = BP/BC * AC + PC

Since Q is the midpoint of AB (as proved in part a), we know that AQ = QB, and therefore:

AP = 1/2 AB

BP = 1/2 AB

Substitute these values into the equation:

AC = (1/2 AB) / BC * AC + PC

Now, let's isolate PC:

PC = AC - (1/2 AB) / BC * AC

PC = AC * (1 - 1/2) / BC

PC = AC * 1/2 / BC

Now, we have the relationship between PC, AC, and BC. Since BC = 1/2 AC (P is the midpoint of AC), we can substitute this into the equation:

PC = AC * 1/2 / (1/2 AC)

PC = AC * 1/2 / 1/2 AC

PC = AC * 1/2 * 2/1 AC

PC = AC * AC / AC

PC = AC (Cancel out AC)

So, we have PC = AC, which means that PA = PC. Therefore, PB = PA = 1/2 AC.

This completes the proof for both parts a) and b).

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