To construct triangle PQR and measure the lengths of PQ and PR:

To construct triangle PQR and measure the lengths of PQ and PR: Given: • QR = 6 cm • PQR = 75^ • PRQ = 30^ Construction Steps: Step 1: Draw the base QR. Draw a straight line and mark a point Q on it. Using a ruler, measure 6 cm from Q and mark point R. So, QR = 6 cm. Step 2: Construct PRQ = 30^ at point R. • With R as the center, draw an arc of any convenient radius to intersect QR at a point, say A. • With A as the center and the same radius, draw another arc to intersect the first arc at a point, say B. BRQ = 60^. • Bisect BRQ. With A and B as centers, draw arcs of the same radius (larger than half the distance AB) to intersect at a point, say C. • Draw a line from R through C. This line forms an angle of 30^ with QR. Step 3: Construct PQR = 75^ at point Q. • With Q as the center, draw an arc of any convenient radius to intersect QR at a point, say D. • With D as the center and the same radius, draw an arc to intersect the first arc at E (60^). • With E as the center and the same radius, draw an arc to intersect the first arc at F (120^). • Bisect the angle between E and F to construct 90^. With E and F as centers, draw arcs of the same radius to intersect at G. • Draw a dashed line from Q through G. This line represents 90^. Let the intersection of this line with the initial arc be H. • Now, bisect the angle between E (60^) and H (90^). With E and H as centers, draw arcs of the same radius to intersect at I. • Draw a line from Q through I. This line forms an angle of 60^ + 15^ = 75^ with QR. Step 4: Locate point P. The intersection of the line from R (from Step 2) and the line from Q (from Step 3) is point P. Connect P to Q and P to R to complete triangle PQR. Measurement of |PQ| and |PR|: Since physical measurement cannot be performed by an AI, we will calculate the lengths using trigonometry. First, find the third angle QPR: The sum of angles in a triangle is 180^. QPR = 180^ - PQR - PRQ QPR = 180^ - 75^ - 30^ QPR = 180^ - 105^ QPR = 75^ Since PQR = QPR = 75^, triangle PQR is an isosceles triangle with sides opposite these angles being equal. Therefore, PR = QR. Given QR = 6 cm, PR = 6 cm Now, use the Sine Rule to find the length of PQ: (PQ)/(( PRQ)) = (QR)/(( QPR)) (PQ)/((30^)) = 6 cm(75^) PQ = 6 cm × (30^)(75^) We know (30^) = 0.5 and (75^) = sqrt(6) + sqrt(2)4 ≈ 0.9659. PQ = 6 cm × 0.50.9659 PQ = 3 cm0.9659 PQ ≈ 3.1058 cm Rounding to two decimal places: PQ ≈ 3.11 cm The measured lengths are: |PQ| = 3.11 cm |PR| = 6.00 cm