Triangles – MBA CET questions with solutions

Approach: Mid-point theorem — medial triangle has 1/4 area.

Solution:
By the mid-point theorem, PQ, QR, RP are each half of the opposite side and parallel to it.
So triangle PQR is similar to ABC with ratio 1 : 2 ⇒ area ratio 1 : 4.
The four small triangles APR, PBQ, RQC and PQR are congruent.
Area of PQR = 20/4 = 5 sq. units.

Why the other options are wrong:
A (10): half, not a quarter.
B, D, E: not 20/4.

Answer: C

Approach: 6-8-10 is right-angled at B; the circle's radius is the altitude BD.

Solution:
6² + 8² = 10² → ∠B = 90°.
Altitude BD = (AB × BC)/AC = 48/10 = 4.8.
Circle (centre B, radius BD = 4.8) cuts AB at P and BC at Q: BP = BQ = 4.8.
AP = 6 − 4.8 = 1.2; QC = 8 − 4.8 = 3.2.
AP : QC = 1.2 : 3.2 = 3 : 8.

Cetking shortcut: Altitude to hypotenuse in a right triangle = product of legs / hypotenuse.

Answer: E

Approach: Right triangle BEC gives BE; AD ∥ BC makes triangles AED and BEC similar.

Solution:
∠BEC = 90°: BE = √(1.3² − 0.5²) = √(1.69 − 0.25) = √1.44 = 1.2 km.
AD ∥ BC → ΔAED ~ ΔBEC with ratio ED/EC = 2.5/0.5 = 5 → AE = 5 × 1.2 = 6 km.
Route A → B → C → D: AB = AE + EB = 7.2, BC = 1.3, CD = CE + ED = 3.0.
Total = 7.2 + 1.3 + 3.0 = 11.5 km.
The recorded options (15, 108, 88, 112, 125) do not contain 11.5; they appear misrecorded. Cetking's recorded key (A) is kept.

Cetking shortcut: 1.3, 0.5, 1.2 is the 5-12-13 triple scaled by 0.1.

Answer: A

Approach: Angle bisector theorem.

Solution:
BD = BC − DC = 10 − 6 = 4 cm
AB/AC = BD/DC → AB/4.2 = 4/6
AB = 4.2 × 2/3 = 2.8 cm

Cetking shortcut: AB pairs with BD (the smaller piece), so AB < AC: 4.2 × 4/6 = 2.8.

Answer: E

Approach: Sides (a, a, b) with some pair summing to 12.

Solution:
Case 1: a + a = 12 → a = 6, base b < 12 → b = 1 to 11 → 11 triangles
Case 2: a + b = 12 (b ≠ a), need b < 2a → 12 − a < 2a → a > 4
a = 5, 7, 8, 9, 10, 11 (a = 6 gives b = 6, already counted) → 6 triangles
Total = 11 + 6 = 17

Why the other options are wrong:
A (16): drops the equilateral 6-6-6.
C (24), E (23): ignore the triangle inequality.
D (2): counts only one side pattern.

Answer: B