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In a right angled triangle, the angle opposite one of the legs is $60^
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11 Aug 2025, 10:12
11 Aug 2025, 10:12
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In a right angled triangle, the angle opposite one of the legs is $\(60°\)$. Which of the following will be independently sufficient to find the area of the triangle?
Indicate all that apply
\(\boxed{A}\) perimeter of the triangle
\(\boxed{B}\) length of the hypotenuse
\(\boxed{C}\) the ratio of the sides opposite angles $\(30^{\circ}\)$ and $\(60^{\circ}\)$ is $\(1: \sqrt{3}\)$
Indicate all that apply
\(\boxed{A}\) perimeter of the triangle
\(\boxed{B}\) length of the hypotenuse
\(\boxed{C}\) the ratio of the sides opposite angles $\(30^{\circ}\)$ and $\(60^{\circ}\)$ is $\(1: \sqrt{3}\)$
Re: In a right angled triangle, the angle opposite one of the legs is $60^
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12 Aug 2025, 11:25
12 Aug 2025, 11:25
Hi Carcass,
Hope you're doing well.
Can you please elaborate on why the answer for this question is 'A' and 'B' instead of 'B' and 'C'?
My Logic/Thinking process for this question:
"In a right angled triangle" -> means largest angle of the triangle will be 90°. And it is also mentioned in the question that angles opposite to one of the legs is 60°. As the sum of the angles of triangle equals to 180°, so the remaining angle of the triangle, opposite of another or smaller leg would be 30°.
So, now we know that the triangle is a 30-60-90 triangle and the ratio of it's sides is 1:√3:2
Option 'A': Just wanted to confirm that are we thinking in this way (I hadn't marked this option earlier because I thought it lacked required information and it cannot be determined) ->
As we know the ratio of the side of the triangle i.e 1:√3:2,
Let x be the length of the smallest side of the triangle.
Then,
Length of side opposite to 60° will be '√3x'
Length of side opposite to 90° will be '2x'
So,
1x+√3x+ 2x = perimeter of the triangle
From here we get the length of the smallest side of the triangle and accordingly we can calculate the other sides as well.
So, now we can calculate the area of the triangle as we have the sides of the triangle. Is that correct?
Option 'B': Length of hypotenuse can be determined with same approach mentioned in Option 'A' explanation.
Option 'C': Why this option is wrong? The ratio of the sides opposite angles of 30° and 60°, will always be 1:√3. Please correct me, if I'm wrong here.
Regards,
Prosperous1
Hope you're doing well.
Can you please elaborate on why the answer for this question is 'A' and 'B' instead of 'B' and 'C'?
My Logic/Thinking process for this question:
"In a right angled triangle" -> means largest angle of the triangle will be 90°. And it is also mentioned in the question that angles opposite to one of the legs is 60°. As the sum of the angles of triangle equals to 180°, so the remaining angle of the triangle, opposite of another or smaller leg would be 30°.
So, now we know that the triangle is a 30-60-90 triangle and the ratio of it's sides is 1:√3:2
Option 'A': Just wanted to confirm that are we thinking in this way (I hadn't marked this option earlier because I thought it lacked required information and it cannot be determined) ->
As we know the ratio of the side of the triangle i.e 1:√3:2,
Let x be the length of the smallest side of the triangle.
Then,
Length of side opposite to 60° will be '√3x'
Length of side opposite to 90° will be '2x'
So,
1x+√3x+ 2x = perimeter of the triangle
From here we get the length of the smallest side of the triangle and accordingly we can calculate the other sides as well.
So, now we can calculate the area of the triangle as we have the sides of the triangle. Is that correct?
Option 'B': Length of hypotenuse can be determined with same approach mentioned in Option 'A' explanation.
Option 'C': Why this option is wrong? The ratio of the sides opposite angles of 30° and 60°, will always be 1:√3. Please correct me, if I'm wrong here.
Regards,
Prosperous1
In a right angled triangle, the angle opposite one of the legs is $60^
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12 Aug 2025, 22:46
12 Aug 2025, 22:46
Expert Reply
We are given that one of the angles of a right triangle is 60 degrees, so we get the third angle as $\(180^{\circ}-\left(90^{\circ}+60^{\circ}\right)=30^{\circ}\)$ (Sum of the angles of a triangle is 180 degrees)
GRE We are given that one of the angles of a right.png [ 42.93 KiB | Viewed 11 times ]
We know that in a triangle having angles $\(90^{\circ}, 60^{\circ} \& 30^{\circ}\)$, the corresponding opposite sides are \(2a,a \sqrt{3},\) and \(a\) respectively, where a is any positive real number (As shown in the figure above).
Now, checking from the options, we have
(A) Perimeter of the triangle - which is as per the figure is $\(2 a+a \sqrt{3}+a=3 a+a \sqrt{3}\)$, so if the value of the perimeter of the triangle is known the value of a can be found and if the value of $\(a\)$ is known we can find the area of the triangle. Hence statement (A) is sufficient.
(B) Length of the hypotenuse - which is as per the figure is \(2 a \), so if the value of the hypotenuse is known we will get the value of $a$ and if the value of $a$ is known we can find the area of the triangle. Hence statement (B) is sufficient.
(C) the ratio of the sides opposite angles $\(30^{\circ} \& 60^{\circ}\)$ is $\(1: \sqrt{3}\)$ - which is not sufficient to calculate the length of the sides of the triangle, so using this we cannot find the area of the triangle. Hence statement (C) is insufficient.
Hence statements (A) & (B) are sufficient independently, so are the correct answers.
Attachment:
GRE We are given that one of the angles of a right.png [ 42.93 KiB | Viewed 11 times ]
We know that in a triangle having angles $\(90^{\circ}, 60^{\circ} \& 30^{\circ}\)$, the corresponding opposite sides are \(2a,a \sqrt{3},\) and \(a\) respectively, where a is any positive real number (As shown in the figure above).
Now, checking from the options, we have
(A) Perimeter of the triangle - which is as per the figure is $\(2 a+a \sqrt{3}+a=3 a+a \sqrt{3}\)$, so if the value of the perimeter of the triangle is known the value of a can be found and if the value of $\(a\)$ is known we can find the area of the triangle. Hence statement (A) is sufficient.
(B) Length of the hypotenuse - which is as per the figure is \(2 a \), so if the value of the hypotenuse is known we will get the value of $a$ and if the value of $a$ is known we can find the area of the triangle. Hence statement (B) is sufficient.
(C) the ratio of the sides opposite angles $\(30^{\circ} \& 60^{\circ}\)$ is $\(1: \sqrt{3}\)$ - which is not sufficient to calculate the length of the sides of the triangle, so using this we cannot find the area of the triangle. Hence statement (C) is insufficient.
Hence statements (A) & (B) are sufficient independently, so are the correct answers.
