Triangle PQR is inscribed in a circle of radius 14 cm. Side PQ is the diameter of the circle and side PR is 10 cm long. What is the area, in square centimetres, of triangle PQR?

Introduction / Context:
This geometry problem combines properties of circles and triangles. When a triangle is inscribed in a circle and one of its sides is a diameter, the triangle is right angled. Knowing this allows us to apply the right triangle area formula. The question provides the radius of the circle and the length of one side, and asks for the area of the entire triangle.

Given Data / Assumptions:

• The circle has radius 14 cm, so its diameter PQ is 28 cm.
• Triangle PQR is inscribed in this circle.
• PQ is the diameter and PR = 10 cm.
• We must find the area of triangle PQR in square centimetres.
• Standard Euclidean geometry rules apply.

Concept / Approach:
By Thales theorem, any triangle inscribed in a circle with a diameter as one side is a right angled triangle, with the right angle at the vertex opposite the diameter. Here, PQ is the diameter, so angle ∠PRQ is 90°. That means triangle PQR is right angled at R. We then know two sides: PR and PQ, and we can compute the third side QR using the Pythagoras theorem. After that, the area of the right triangle is: Area = (1/2) * (leg 1) * (leg 2) where the legs are PR and QR.

Step-by-Step Solution:
Step 1: Use the fact that PQ is the diameter of the circle. Radius r = 14 cm, so diameter PQ = 2 * 14 = 28 cm. Step 2: Since PQ is the diameter, angle ∠PRQ is 90°, so PQR is a right triangle with right angle at R. Step 3: Apply the Pythagoras theorem to find QR. Let PR = 10 cm, PQ = 28 cm and QR be the unknown side. PQ^2 = PR^2 + QR^2 28^2 = 10^2 + QR^2 784 = 100 + QR^2 QR^2 = 784 − 100 = 684 QR = √684 = √(4 * 171) = 2√171 Step 4: Compute the area of the right triangle using legs PR and QR. Area = (1/2) * PR * QR = (1/2) * 10 * 2√171 = 10√171 Step 5: Simplify √171. Notice that 171 = 9 * 19, so √171 = √(9 * 19) = 3√19. Thus, Area = 10 * 3√19 = 30√19 square centimetres.

Verification / Alternative check:
We can approximate the answer numerically to verify reasonableness. Take √19 ≈ 4.36. Then 30√19 ≈ 30 * 4.36 ≈ 130.8 square centimetres. Using legs PR = 10 and QR ≈ 2 * √171 ≈ 2 * 13.08 ≈ 26.16, the area (1/2) * 10 * 26.16 ≈ 130.8 square centimetres, which matches the simplified surd value 30√19. This confirms that the symbolic and approximate calculations agree.

Why Other Options Are Wrong:
Option 1: 196 sq cm would correspond to a triangle with much larger legs; it does not match the computed area using the given side lengths. Option 3: 40√17 sq cm is based on a different surd and does not come from the correct Pythagoras relation for this triangle. Option 4: 35√21 sq cm again uses unrelated numbers; substituting the correct side lengths never yields this value. Option 5: None of these is wrong because 30√19 sq cm is a listed option and matches the correct calculation.

Common Pitfalls:
A typical error is forgetting that the triangle is right angled because PQ is a diameter. Some learners may attempt to use a general triangle area formula with circumradius, which is more complicated and error prone. Others might incorrectly treat the radius as 10 cm or 20 cm instead of the given 14 cm, which changes the diameter and invalidates the right angle conclusion. Also, handling surds carelessly when simplifying √684 can lead to mistakes; recognizing 684 as 4 * 171 and then 171 as 9 * 19 is crucial for simplifying the root correctly.

Final Answer:
The area of triangle PQR is 30√19 sq cm.