Answer:
The answer is below.
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