Can someone please help me with the following problems
Determine if the triple numbers are can be sides of a right triangle?
sqrt(10)sqrt(13)sqrt(23)
The length of thy hypotenuse of a right triangle equals the length of one of legs of the triangle
A) always true
B) sometimes true
C) never true
Do the followig figures form semiregular tesselations?
Pentagon, hexagon 15-gon
A)yes
B)NO
Is √23^2 = √10^2 + √13^2 ?
How can the hypotenuse , which is always the "longest" side be equal to one of the others ?
Sure, I'd be happy to help you with these problems!
1. To determine if the triple numbers can be the sides of a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In this case, we have the side lengths as sqrt(10), sqrt(13), and sqrt(23). So, we can check if these satisfy the Pythagorean theorem by squaring each of them and seeing if the sum of the squares of two of the sides is equal to the square of the remaining side.
Let's calculate:
- Squaring sqrt(10): (sqrt(10))^2 = 10
- Squaring sqrt(13): (sqrt(13))^2 = 13
- Squaring sqrt(23): (sqrt(23))^2 = 23
Now, let's check if the sum of the squares of two sides is equal to the square of the remaining side:
- 10 + 13 = 23, but sqrt(23) is not equal to 23
Since the sum of the squares of two sides is not equal to the square of the remaining side, these triple numbers cannot be the sides of a right triangle.
2. The statement "The length of the hypotenuse of a right triangle equals the length of one of the legs of the triangle" is sometimes true. In a right triangle, the hypotenuse is the side opposite to the right angle, and the other two sides are called the legs. If one of the legs has the same length as the hypotenuse, then it forms a special right triangle known as an isosceles right triangle. So, the answer is B) sometimes true.
3. To determine if the given figures form semiregular tessellations, we need to check if each vertex of the figures has an equal number of edges meeting. In a semiregular tessellation, each regular polygon should share a vertex with the same number of other polygons.
The given figures are a pentagon, a hexagon, and a 15-gon.
For the pentagon, each vertex has 3 edges meeting: A pentagon is regular and has 5 sides, so each vertex will have 5 edges divided into 5 equal angles. Therefore, it does not form a semiregular tessellation.
For the hexagon, each vertex has 3 edges meeting: A hexagon is regular and has 6 sides, so each vertex will have 6 edges divided into 6 equal angles. Therefore, it forms a semiregular tessellation.
For the 15-gon, each vertex has 4 edges meeting: A 15-gon does not have equal angles for each side. Therefore, it does not form a semiregular tessellation.
Based on the above analysis, the answer is B) NO, only the hexagon forms a semiregular tessellation.
I hope this helps! Let me know if you have any further questions.