2. ABCD is a square and P, Q are the midpoints of BC, CD respectively. If AP = a and AQ = b, find in terms of a and b, the directed line segments (i) AB, (ii) AD, (iii) BD and (iv) AC.

The answer is below.

Step-by-step explanation:

A square is a quadrilateral (has four sides and four angles) in which all the sides are equal. Also all the angle of a square are equal and measure 90°. The diagonals of a square are equal.

In square ABCD:

AB = BC = CD = AC

Given that  P, Q are the midpoints of BC, CD respectively. If AP = a and AQ = b.

a) In triangle ABP:

BC = AB (all sides of a triangle are equal)

BP = 1/2 (BC) = 1/2(AB) = 0.5AB

Using Pythagoras theorem:

AB² + BP² = AP²

substituting:

AB² + (0.5AB)² = a²

AB² + 0.25AB² = a²

1.25AB² = a²

AB² = a² / 1.25

AB² = 0.8a²

AB = √(0.8a²)

AB = a√0.8

b) In triangle ADQ:

AD = CD (all sides of a triangle are equal)

DQ = 1/2 (CD) = 1/2(AD) = 0.5AD

Using Pythagoras theorem:

AD² + DQ² = AQ²

substituting:

AD² + (0.5AD)² = b²

AD² + 0.25AD² = b²

1.25AD² = b²

AD² = b² / 1.25

AD² = 0.8b²

AD = √(0.8b²)

AD = b√0.8

c) In triangle ABD, Using Pythagoras theorem:

AB² + AD² = BD²

Substituting:

0.8a² + 0.8b² = BD²

But AB = AD (all sides of a triangle are equal). Hence AB = AD = 0.8a² = 0.8b²

BD = √(0.8a² + 0.8b²)

BD = √(1.6a²)     0.8a² = 0.8b²

BD = a√1.6

D) AC = AD (diagonal are equal)

AC = √(0.8a² + 0.8b²)

AC = √(1.6b²)     0.8a² = 0.8b²

AC = b√1.6

constant of proportionality = 3 and the equation is c = 3p.

step-by-step explanation:

c = kp   where c = amount of caffeine in mg., p = oz of pepsi and k is the constant of proportionality.

substituting the values given:

24 = 8k

k = 3.

the same value of k will be obtained if we substitute the other given values.