Constructing triangle PQR.

a) Step 1: Constructing triangle PQR. • Draw a line segment PQ of length 8 cm. • At point P, construct an angle of 45^ with PQ. Draw a ray from P. • Since QRP = 90^ and QPR = 45^, the third angle PQR = 180^ - 90^ - 45^ = 45^. • At point Q, construct an angle of 45^ with PQ. Draw a ray from Q. • The intersection of the two rays from P and Q is point R. • Measure the length of QR. (This measurement cannot be provided without physical construction). Step 2: Constructing the mediator of PQ and finding S. • With P as the center and a radius greater than half the length of PQ (i.e., greater than 4 cm), draw arcs above and below PQ. • With Q as the center and the same radius, draw arcs that intersect the previously drawn arcs. • Draw a straight line connecting the two intersection points of the arcs. This line is the perpendicular bisector (mediator) of PQ. • Mark the point where this perpendicular bisector intersects the line segment PR as S. Step 3: Constructing a circle. • Place the compass point at S (the center). • Adjust the compass opening to a radius of 3 cm. • Draw a circle with center S and radius 3 cm. b) Step 1: Identify the components of the right-angled triangle formed by the broken tree. The height where the tree broke is one leg of the right triangle: h_1 = 12 ft. The distance from the base of the tree to where the top touches the ground is the other leg: d = 13 ft. The broken part of the tree forms the hypotenuse: h_2. Step 2: Use the Pythagorean theorem to find the length of the broken part (h_2). h_1^2 + d^2 = h_2^2 (12 ft)^2 + (13 ft)^2 = h_2^2 144 ft^2 + 169 ft^2 = h_2^2 313 ft^2 = h_2^2 h_2 = sqrt(313) ft Step 3: Calculate the total height of the tree before it broke. The total height H is the sum of the standing part and the broken part. H = h_1 + h_2 H = 12 ft + sqrt(313) ft H ≈ 12 ft + 17.6918 ft H ≈ 29.69 ft The height of the tree before the wind broke it was 12 + sqrt(313) ft ≈ 29.69 ft. c.i) Step 1: Write down the expression and given values. Expression: (8v^2 + 2vu)/(7r) × (14r^2)/(4v+u) Values: v=2, u=-2, r=3 Step 2: Simplify the expression by factoring. Factor out 2v from the numerator of the first fraction: 8v^2 + 2vu = 2v(4v+u) Substitute this back into the expression: (2v(4v+u))/(7r) × (14r^2)/(4v+u) Cancel out the common term (4v+u): (2v)/(7r) × 14r^2 Simplify the constants and r terms: (2v × 14r^2)/(7r) = (2v × (2 × 7) × r × r)/(7r) Cancel out 7r: 2v × 2r = 4vr Step 3: Substitute the given values into the simplified expression. 4vr = 4(2)(3) 4(2)(3) = 8(3) 8(3) = 24 The evaluated expression is 24. c.ii) Step 1: Write down the expression to factorize. 12v^3 - 2u^2 + 8v^2u - 3vu Step 2: Rearrange terms for grouping. Group terms with common factors: (12v^3 + 8v^2u) - (3vu + 2u^2) Step 3: Factor out common terms from each group. From the first group (12v^3 + 8v^2u), the common factor is 4v^2: 4v^2(3v + 2u) From the second group (-3vu - 2u^2), the common factor is -u: -u(3v + 2u) Step 4: Combine the factored groups. (4v^2(3v + 2u)) - u(3v + 2u) Now, (3v + 2u) is a common factor for both terms: (3v + 2u)(4v^2 - u) The factored expression is (3v + 2u)(4v^2 - u).