Tell weather a triangle with side lengths of 10,24,26 could be 45 45 90 degrees or 30 60 90 degrees. Explain.
the first question is: Is this a right triangle?
5 12 13 are half the sides
is 25 + 144 = 169 ?
yes, so it is a right triangle
now could it be 45 , 45 , 90 ?
No, because sides opposite congruent angles are equal. Our triangle has no two equal length sides.
Now, could it be 30, 60, 90 ?
Well, if it were then the hypotenuse would be twice the shortest side because sin 30 = 1/2
Is 24 twice 10 ?
No, so it is not 30, 60, 90 either.
I mean is 26 twice 10 ?
To determine whether a triangle with side lengths of 10, 24, and 26 could be a 45-45-90 or a 30-60-90 triangle, we need to compare the given side lengths with the ratios associated with each triangle type.
Let's start with the 45-45-90 triangle. In a 45-45-90 triangle, the two legs (the sides opposite the 45-degree angles) are congruent, and the length of the hypotenuse (the side opposite the right angle) can be found using the following ratio:
leg length : leg length : hypotenuse length
= 1 : 1 : √2
Now, let's use this ratio to check if the given side lengths of 10, 24, and 26 match the ratios of a 45-45-90 triangle.
For a 45-45-90 triangle, the two leg lengths should be equal. However, in this case, the side lengths 10 and 24 are not equal, so the given triangle cannot be a 45-45-90 triangle.
Next, let's consider a 30-60-90 triangle. In a 30-60-90 triangle, the sides are in the following ratio:
shorter leg : longer leg : hypotenuse
= 1 : √3 : 2
Again, let's check if the given side lengths of 10, 24, and 26 match the ratios of a 30-60-90 triangle.
We can see that the side length 10 is equal to the shorter leg, and the side length 26 is equal to the hypotenuse. The ratio between the shorter leg and the hypotenuse is 1 : 2, which is consistent with a 30-60-90 triangle.
However, to confirm if it is indeed a 30-60-90 triangle, we need to find the length for the longer leg using the ratio 1 : √3 : 2.
length of longer leg = (length of shorter leg) * √3
= 10 * √3
≈ 17.32
The length of the longer leg in the given triangle is 24, which is not equal to 17.32. Therefore, the given triangle cannot be a 30-60-90 triangle either.
In conclusion, the triangle with side lengths of 10, 24, and 26 cannot be classified as either a 45-45-90 triangle or a 30-60-90 triangle.