Answer :
To solve this problem, let's analyze the given side ratios of the triangle, which are 13:12:5. Our goal is to determine if this triangle is a right-angled triangle and to identify the right angle.
### Step-by-Step Solution:
1. Assign Variables:
Let the sides of the triangle be [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. Given the ratios, we can express these sides as:
[tex]\[ a = 13k, \quad b = 12k, \quad c = 5k \][/tex]
where [tex]\( k \)[/tex] is a positive scaling factor.
2. Pythagorean Theorem:
For a triangle to be a right-angled triangle, the sides must satisfy the Pythagorean theorem (i.e., the square of the length of the longest side is equal to the sum of the squares of the other two sides).
3. Identify the Longest Side:
Since 13 is the largest ratio, [tex]\( a = 13k \)[/tex] would be the hypotenuse if the triangle is right-angled. Thus, we need to check if:
[tex]\[ a^2 = b^2 + c^2 \][/tex]
4. Calculate Each Side Squared:
[tex]\[ a^2 = (13k)^2 = 169k^2 \][/tex]
[tex]\[ b^2 = (12k)^2 = 144k^2 \][/tex]
[tex]\[ c^2 = (5k)^2 = 25k^2 \][/tex]
5. Sum of [tex]\( b^2 \)[/tex] and [tex]\( c^2 \)[/tex]:
[tex]\[ b^2 + c^2 = 144k^2 + 25k^2 = 169k^2 \][/tex]
6. Compare with [tex]\( a^2 \)[/tex]:
[tex]\[ a^2 = 169k^2 \][/tex]
Thus,
[tex]\[ b^2 + c^2 = a^2 \][/tex]
[tex]\[ 169k^2 = 169k^2 \][/tex]
Since the equation holds true, we have verified that the triangle with sides in the ratio 13:12:5 satisfies the Pythagorean theorem.
### Conclusion:
- Proof of Right-Angled Triangle:
Yes, the triangle is a right-angled triangle.
- Right Angle Identification:
Since [tex]\( a^2 = b^2 + c^2 \)[/tex], the right angle is opposite the side with length [tex]\( a \)[/tex]. Therefore, [tex]\( a \)[/tex], which corresponds to the side with the ratio 13, is the hypotenuse, making the angle opposite to [tex]\( a \)[/tex] (13k) the right angle.
In summary:
- The triangle with side ratios 13:12:5 is a right-angled triangle.
- The right angle is opposite the side with length 13k (i.e., side [tex]\( a \)[/tex]).
### Step-by-Step Solution:
1. Assign Variables:
Let the sides of the triangle be [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. Given the ratios, we can express these sides as:
[tex]\[ a = 13k, \quad b = 12k, \quad c = 5k \][/tex]
where [tex]\( k \)[/tex] is a positive scaling factor.
2. Pythagorean Theorem:
For a triangle to be a right-angled triangle, the sides must satisfy the Pythagorean theorem (i.e., the square of the length of the longest side is equal to the sum of the squares of the other two sides).
3. Identify the Longest Side:
Since 13 is the largest ratio, [tex]\( a = 13k \)[/tex] would be the hypotenuse if the triangle is right-angled. Thus, we need to check if:
[tex]\[ a^2 = b^2 + c^2 \][/tex]
4. Calculate Each Side Squared:
[tex]\[ a^2 = (13k)^2 = 169k^2 \][/tex]
[tex]\[ b^2 = (12k)^2 = 144k^2 \][/tex]
[tex]\[ c^2 = (5k)^2 = 25k^2 \][/tex]
5. Sum of [tex]\( b^2 \)[/tex] and [tex]\( c^2 \)[/tex]:
[tex]\[ b^2 + c^2 = 144k^2 + 25k^2 = 169k^2 \][/tex]
6. Compare with [tex]\( a^2 \)[/tex]:
[tex]\[ a^2 = 169k^2 \][/tex]
Thus,
[tex]\[ b^2 + c^2 = a^2 \][/tex]
[tex]\[ 169k^2 = 169k^2 \][/tex]
Since the equation holds true, we have verified that the triangle with sides in the ratio 13:12:5 satisfies the Pythagorean theorem.
### Conclusion:
- Proof of Right-Angled Triangle:
Yes, the triangle is a right-angled triangle.
- Right Angle Identification:
Since [tex]\( a^2 = b^2 + c^2 \)[/tex], the right angle is opposite the side with length [tex]\( a \)[/tex]. Therefore, [tex]\( a \)[/tex], which corresponds to the side with the ratio 13, is the hypotenuse, making the angle opposite to [tex]\( a \)[/tex] (13k) the right angle.
In summary:
- The triangle with side ratios 13:12:5 is a right-angled triangle.
- The right angle is opposite the side with length 13k (i.e., side [tex]\( a \)[/tex]).
