In the figure above, is the area of triangular region ABC equal to the area of triangular region DBA ?...

Understanding the Question

We need to determine whether the area of triangle ABC equals the area of triangle DBA.

Given Information

• AC is vertical, CB is horizontal, with \(\angle \mathrm{ACB} = 90°\)
• \(\mathrm{AD} \perp \mathrm{AB}\) (perpendicular at point A)
• Both triangles share side AB

Key Insight for Area Comparison

Since both triangles share the same base AB, they will have equal areas if and only if they have equal heights (perpendicular distances from C and D to line AB).

Therefore, our question becomes: Is the height from C to line AB equal to the height from D to line AB?

Note: The height from D to AB is simply |AD| since \(\mathrm{AD} \perp \mathrm{AB}\).

Analyzing Statement 1

Statement 1: \((\mathrm{AC})^2 = 2(\mathrm{AD})^2\)

This tells us that \(\mathrm{AC} = \sqrt{2} \cdot \mathrm{AD}\).

What This Gives Us

We now know the exact relationship between AC and AD.

What We Still Need

However, we don't know:

• The length of BC
• Where exactly point B is located (other than being on the horizontal line through C)
• How the height from C to AB relates to AD

Testing Different Scenarios

Scenario 1: If BC is relatively short

• Triangle ABC becomes "tall and narrow"
• The height from C to AB will be relatively small
• This height could be less than AD

Scenario 2: If BC is longer

• Triangle ABC becomes more "spread out"
• The height from C to AB increases
• This height could equal or exceed AD

Since we can construct triangles with \(\mathrm{AC} = \sqrt{2} \cdot \mathrm{AD}\) but different values of BC, we get different heights from C to AB. This means different area relationships are possible.

Conclusion for Statement 1

Statement 1 is NOT sufficient because knowing \(\mathrm{AC} = \sqrt{2} \cdot \mathrm{AD}\) doesn't determine the height from C to AB.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Now we analyze Statement 2 independently, forgetting Statement 1 completely.

Statement 2: Triangle ABC is isosceles

This means two sides of triangle ABC are equal.

What This Gives Us

Three possibilities exist:

1. AC = BC (making a right isosceles triangle)
2. AC = AB
3. BC = AB

Why This Isn't Enough

Without knowing:

• Which sides are equal
• Any specific lengths
• Any relationship to triangle DBA or AD

We cannot determine if the areas match. For example:

• If AC = BC = 1 and AD happens to equal \(1/\sqrt{2}\), the areas might be equal
• If AC = BC = 1 and AD = 2, the areas would definitely differ
• Different isosceles configurations give different area relationships

Conclusion for Statement 2

Statement 2 is NOT sufficient because it tells us nothing about the relationship between triangles ABC and DBA.

[STOP - Not Sufficient!] This eliminates choice B (and confirms D is already eliminated).

Combining Both Statements

Combined Information

From both statements together:

• \(\mathrm{AC} = \sqrt{2} \cdot \mathrm{AD}\) (from Statement 1)
• Triangle ABC is isosceles (from Statement 2)
• \(\mathrm{AC} \perp \mathrm{CB}\) (given in the figure)

The Key Deduction

With \(\mathrm{AC} \perp \mathrm{CB}\) and triangle ABC being isosceles, let's determine which configuration is geometrically possible:

• If AC = AB: In a right triangle, this would mean a leg equals the hypotenuse. Impossible!
• If BC = AB: In a right triangle, this would mean a leg equals the hypotenuse. Impossible!
• Therefore, we must have AC = BC

This makes triangle ABC a 45-45-90 right isosceles triangle.

Finding the Heights

For a 45-45-90 triangle with legs AC = BC:

• The height from C to the hypotenuse AB equals \(\mathrm{AC}/\sqrt{2}\)
• This is a standard property of right isosceles triangles

Now, using Statement 1:

• Height from C to AB = \(\mathrm{AC}/\sqrt{2} = (\sqrt{2} \cdot \mathrm{AD})/\sqrt{2} = \mathrm{AD}\)
• Height from D to AB = AD (given that \(\mathrm{AD} \perp \mathrm{AB}\))

The Answer to Our Question

Since both heights equal AD:

• Same base (AB) + Same height (AD) = Equal areas

[STOP - Sufficient!] We've established the areas must be equal.

The Answer: C

Both statements together are sufficient because they uniquely determine that:

1. Triangle ABC must be right isosceles with AC = BC
2. This configuration, combined with \(\mathrm{AC} = \sqrt{2} \cdot \mathrm{AD}\), makes both triangular areas equal

Neither statement alone provides enough information to answer the question.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient.