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

Let's solve this by setting up coordinates instead of the median-length formula, which makes it obvious why both pieces of information are needed together.

Place the right angle at the origin: $B=(0,0)$, $A=(c,0)$ on one leg, and $C=(0,a)$ on the other leg, where $c=AB$ and $a=BC$.

Using Statement (A) alone: D being the midpoint of BC gives $D=(0, a/2)$, and E being the midpoint of AB gives $E=(c/2, 0)$. This is purely a statement about *where* D and E sit; it says nothing about the actual size of the triangle, so a, c, and hence AC, are completely unknown. Not sufficient.

Using Statement (B) alone: we are told two segment lengths, AD = 7 and CE = 5, but without knowing where D and E are located on their sides, these numbers cannot be tied to a and c in any single way. Not sufficient.

Using both together: with $D=(0,a/2)$ and $A=(c,0)$,

$$AD = \sqrt{c^2 + (a/2)^2} = 7 \implies c^2 + \frac{a^2}{4} = 49$$

With $E=(c/2,0)$ and $C=(0,a)$,

$$CE = \sqrt{a^2 + (c/2)^2} = 5 \implies a^2 + \frac{c^2}{4} = 25$$

Adding these two equations:

$$\left(a^2+c^2\right)\left(1+\frac14\right) = 74 \implies a^2+c^2 = \frac{296}{5}$$

Since $AC=\sqrt{a^2+c^2}$ (AC is the hypotenuse, running from $A=(c,0)$ to $C=(0,a)$), we get a single fixed value:

$$AC = \sqrt{\frac{296}{5}} = \frac{2\sqrt{370}}{5} \approx 7.69$$

Let's summarize:

• Statement (A) only fixes the position of D and E relative to the triangle, with no size information.
• Statement (B) only gives two lengths, but they cannot be used without knowing D and E are midpoints.
• Together, the two median-style equations combine to give a single value of $a^2+c^2$, and hence of AC.

So the two statements together are sufficient, but neither one alone is.