PQRS is a square. If x is the midpoint of PQ, calculate correct to the nearest degree, <PXS.

Analyze the question: PQRS is a square and X is the midpoint of PQ. We need the angle PXS, which lies in triangle PXS.

Concept: In a square, each angle is 90°. Triangle PXS is right-angled at P, so we can use the tangent ratio.

Solution:

Let the side of the square be 2a. Then:

• PS = 2a
• PX = a (X is the midpoint of PQ)

In right-angled triangle PXS (right angle at P), for angle PXS:

• Opposite side = PS = 2a
• Adjacent side = PX = a

tan(PXS) = PS / PX = 2a / a = 2

PXS = tan⁻¹(2) ≈ 63.43°

To the nearest degree, PXS ≈ 63°.

Answer: C (63°)

Tip: When a point is the midpoint of a side of a square, the ratio of the sides in the right-angled triangle is 2 : 1, so tan θ = 2. Remember tan⁻¹(2) ≈ 63.4° (and tan⁻¹(1/2) ≈ 26.6°). They add up to 90°, which is a quick way to check your work.