Consider two triangles ABC and ABD with internal angles as shown and a common side AB. What is the length of side AC?(1) AD2+AC2=1000(2) CD=10(3+1)a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'D'. Can you explain this answer?

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Consider two triangles ABC and ABD with internal angles as shown and a common side AB. What is the length of side AC?

(1) AD²+AC²=1000

(2) CD=10(√3+1)

a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
c)
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
d)
EACH statement ALONE is sufficient.
e)
Statements (1) and (2) TOGETHER are NOT sufficient.

Correct answer is option 'D'. Can you explain this answer?

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Consider two triangles ABC and ABD with internal angles as shown and a...

Steps 1 & 2: Understand Question and Draw Inferences

Here we can see that triangles ABC and ABD form the special triangles 30°-60°-90° and 45°-45°-90° respectively.

And we know that:

i) In a 30°-60°-90° triangle, the sides opposite to the angles 30°, 60°, and 90° respectively are in the ratio 1 : √3 : 2

ii) In a 45°-45°-90° triangle, the sides opposite to the angles 45°, 45°, and 90° respectively are in the ratio 1 : 1: √2

Let’s assume the length of side BC to be x units.

The lengths of all the other sides of the two triangles can be calculated in terms of x by applying the above ratios. The result will be as shown:

Thus, in order to find the length of AC, we need to find the value of x.

Step 3: Analyze Statement 1

AD²+AC²=1000

This means

(√6x)²+(2x)²=1000

6x²+4x²=1000

10x²=1000

x²=100

x=10

(The negative value of x is rejected because x, being the length of a geometrical figure, cannot be negative)

Since we know the value of x, we will be able to find the length of AC.

Thus, Statement 1 alone is sufficient.

Step 4: Analyze Statement 2

CD=10(√3+1)

This means,

x+3√x=10(3√+1)

x(1+√3)=10(√3+1)

x=10

Since we know the value of x, we will be able to find the length of AC.

Thus, Statement 2 alone is sufficient

Step 5: Analyze Both Statements Together (if needed)

We do not need to analyze further as both statements (1) and (2) are sufficient.

Answer: Option (D)

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Consider two triangles ABC and ABD with internal angles as shown and a common side AB. What is the length of side AC?(1) AD2+AC2=1000(2) CD=10(√3+1)a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'D'. Can you explain this answer?

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