Question

Directions: Each of the following questions is followed by two statements, labelled (A) and (B). Decide whether the statements are sufficient to conclusively answer the question, and choose:
(A) if Statement (A) alone is sufficient but Statement (B) alone is not.
(B) if Statement (B) alone is sufficient but Statement (A) alone is not.
(C) if Statement (A) and Statement (B) together are sufficient, but neither alone is sufficient.
(D) if either Statement (A) alone or Statement (B) alone is sufficient.
(E) if both statements together are still not sufficient.

ABC is a triangle with \(\angle B = 90^{\circ}\). What is the length of the side AC?
(A) D is the midpoint of BC and E is the midpoint of AB.
(B) AD = 7 and CE = 5.

• (A) Statement (A) alone is sufficient, but Statement (B) alone is not sufficient.
• (B) Statement (B) alone is sufficient, but Statement (A) alone is not sufficient.
• (C) Statement (A) and Statement (B) together are sufficient, but neither alone is sufficient.
• (D) Either Statement (A) alone or Statement (B) alone is sufficient.

Hint:

Write AD and CE as the hypotenuses of two small right triangles formed using the right angle at B; you need both the midpoint fact and the two numeric lengths to get two equations in AB and BC.

Answer 1

The Correct Option is C

Step 1: Understand the question.
Triangle ABC has a right angle at B, so AC is the hypotenuse. We are asked whether we can pin down AC using facts about two cevians, AD and CE.

Step 2: Test Statement A alone.
Statement A only tells us D and E are midpoints of BC and AB, i.e. AD and CE are medians. It gives no lengths at all, so we cannot compute AC. Not sufficient alone.

Step 3: Test Statement B alone.
Statement B gives AD = 7 and CE = 5, but without knowing what D and E actually are (midpoints, or some other points on the sides), these two lengths cannot be tied to AC in any fixed way. Not sufficient alone.

Step 4: Combine both statements.
Let AB = c and BC = a, with the right angle at B. Since D is the midpoint of BC, \(BD = a/2\); since E is the midpoint of AB, \(BE = c/2\). Because angle B is \(90^{\circ}\), triangle ABD and triangle CBE are both right-angled at B, so:
\[ AD^2 = AB^2 + BD^2 = c^2 + \frac{a^2}{4} = 49 \]
\[ CE^2 = CB^2 + BE^2 = a^2 + \frac{c^2}{4} = 25 \]

Step 5: Solve the two equations together.
Adding the two equations:
\[ \left(c^2+a^2\right) + \frac{a^2+c^2}{4} = 74 \implies \frac{5}{4}\left(a^2+c^2\right) = 74 \implies a^2+c^2 = \frac{296}{5} \]
Since AC is the hypotenuse, \(AC^2 = a^2+c^2 = \dfrac{296}{5}\), which gives a single, definite numeric value:
\[ AC = \sqrt{\dfrac{296}{5}} = \dfrac{2\sqrt{370}}{5} \approx 7.69 \]
So once both statements are used together, AC is completely determined. Neither statement was enough by itself.

Final Answer:
\[ \boxed{\text{Statement (A) and (B) together are sufficient, neither alone is}} \]