CMAT questions on Data Sufficiency

Statement II: chords XY and ZT meet at P inside the circle.

• ∠XPZ = ∠TPY, because they are vertically opposite angles.
• ∠PXZ (= ∠YXZ) = ∠PTY (= ∠ZTY), because both stand on arc YZ.

Two angles are equal, so ΔPXZ ~ ΔPTY. True.

Statement I: from this similarity, PX/PT = PZ/PY, so PX · PY = PZ · PT. This is the intersecting-chords theorem. True.

Answer: Both Statement I and Statement II are true (option 1)

Statement II (prove this first). D is any point on BC. Apply the triangle inequality in ΔABD and ΔADC:

• AD < AB + BD
• AD < AC + DC

Adding: 2AD < AB + AC + (BD + DC) = AB + BC + CA. True.

Statement I: apply II to each median in turn (D taken as the midpoint of the opposite side):

• 2m_a < perimeter
• 2m_b < perimeter
• 2m_c < perimeter

Adding gives only m_a + m_b + m_c < 1.5 × perimeter, which is not enough. A sharper standard argument does work: extend each median to double its length to form a parallelogram, which gives 2m_a < b + c. Adding over all three medians gives m_a + m_b + m_c < a + b + c. True. (The sum of the medians lies between 3/4 of the perimeter and the full perimeter.)

Answer: Both Statement I and Statement II are true (option 1)

Asked: which assumptions the statement takes for granted.

Statement: if people ask who your tailor is, your shirt must be excellent. People's inquiry is treated as the sign of an excellent shirt.

Assumption I: people do not ask about the tailor when the shirt is not good. This must be assumed. If people asked about the tailor even for poor shirts, their asking could not prove the shirt is excellent. I is implicit.

Assumption II: people want to know the criteria of an excellent shirt. The statement is about people wanting to find the tailor, not about wanting to learn what makes a shirt excellent. Nothing in it needs II. II is not implicit.

Answer: Only Assumption I is implicit