In right triangle ABC above, BC = 12, and AD = 9. Which of the followi

In right triangle ABCABC above, BC=12, and AD=9. Which of the following statements, considered individually, is sufficient to determine the triangle’s area?

Area of triangle ABC = 1/2 * AB* BC, we have BC known, we need to know AB to get area of triangle ABC.

I. Triangle ABD is equilateral -> AD= AB = 9, so sufficient to get area of triangle ABC.

II. DC=6 -> AC = AD +DC = 9+6 = 15, So, we can get, AB =Sqrt (AC^2 - BC^2). Sufficient to get area of triangle ABC.

III. AB=3/4 *BC -> So, AB can be calculated. And, now, Sufficient to get area of triangle ABC.

So, I think E.

imo E .

For area we need base and height . Base we know . lets evaluate for height AB .

I. Triangle \(ABD\) is equilateral so AB = AD .. sufficient .
II . \(DC = 6\) So .. AC can be found => AB can be found .. Sufficient .
III. \(AB=\frac{3}{4}BC\) => AB can be found ..

So ans is E .

Given

• ABC is a right-angle triangle.
• BC=12 and AD=9

To find

• Statement that is/are sufficient to find area of ABC.

Approach

Statement I: Triangle ABD is equilateral

• AD = AB = 9

o We can now find the area using ½ × BC ×AB.

Statement II: DC=6

• We can find length of AD as AD = DC + AD

o Then using pythagorus theorem, we can find AB and then using area formula, we can find the answer.

Statement III: AB=3/4BC

• We already know the value of BC. So, we can find AB.
• Then, using the area formula, we can find the area.

Hence, the correct answer is option E.

Statement 1. If it is equilateral, then all sides will be 9, thus suffice
2: DC = 6, then we can use phyth. theorem to determine; suffice
3: AB= 3/4BC, AB= 9; suffice

Thus all suffice

Triangle ABC is a right angled triangle, right angled at B. AC is the hypotenuse and hence,

\(AC^2\) = \(AB^2\) + \(BC^2\).

It is also known that BC = 12 and AD = 9.

Time to interpret the statements individually to evaluate their sufficiency to find the area of triangle ABC.

Area of triangle ABC = ½ * AB * BC = ½ * 12 * AB.
Therefore, any statement that can help us find AB is sufficient on its own.

Statement I: Triangle ABD is equilateral.

Therefore, AB = AD = 9 cm.
Statement I alone is sufficient. Answer options B and C can be eliminated.

Statement II: DC = 6

From the question, AD = 9 and also \(AC^2\) = \(BC^2\) + \(AB^2\)

From the diagram, AC = AD + DC; therefore, AC = 15. We know that BC = 12.
Substituting in the equation, AB can be calculated.
Statement II alone is sufficient. Answer option A can be eliminated.

Statement III: AB = ¾ BC.

Since BC = 12, the value of AB can be calculated.
Statement III alone is sufficient. Answer option D can be eliminated.

The correct answer option is E.