Respuesta :
Answer:
Set B i.e., 5 , 7 , 8 represent the side of acute angled triangle.
Step-by-step explanation:
Given: Set of numbers.
To find: Set that represent sides of an acute angled traingle.
We use the following result:
When given 3 triangle sides then to determine if the triangle is acute angled , right angled or obtuse angled.
First find Square all 3 sides, then Sum the squares of the 2 shortest sides and then Compare the sum to the square of the last side.
if sum > Square of last side ⇒ it is Acute Triangle
if sum = Square of last side ⇒ it is Right Triangle
if sum < Square of last side ⇒ it is Obtuse Triangle
a). 4 , 5 , 7
4² = 16 , 5² = 25 , 7² = 49
16 + 25 = 41
∵ 41 < 49
⇒ It is an Obtuse Traingle.
b). 5 , 7 , 8
5² = 25 , 7² = 49 , 8² = 64
25 + 49 = 74
∵ 74 > 64
⇒ It is an acute Triangle.
c). 6 , 7 , 10
6² = 36 , 7² = 49 , 10² = 100
36 + 49 = 85
∵ 85 < 100
⇒ It is an Obtuse Triangle.
d). 7 , 9 , 12
7² = 49 , 9² = 81 , 12² = 144
49 + 81 = 130
∵ 130 < 144
⇒ It is an Obtuse Triangle.
Therefore, Set B i.e., 5 , 7 , 8 represent the side of acute angled triangle.
The set of number which represent the length of the sides of an acute triangle is [tex]5,7,8[/tex] i.e., [tex]\fbox{\begin\\\ \bf option 2\\\end{minispace}}[/tex].
Further explanation:
A triangle is two dimensional figure which is formed when three non-collinear points are joined. It has three sides, three angles and three edges.
Consider a triangle with sides as [tex]L_{1},L_{2} \text{and} L_{3}[/tex] such that [tex]L_{3}>L_{2}>L_{1}[/tex].
The categorization of a triangle on the basis of side is as follows:
1) Acute triangle: If the square of the longest side is smaller than the sum of the square of the other two sides than the triangle is an acute triangle.
This implies that if, [tex](L_{3})^{2}<(L_{1})^{2}+(L_{2})^{2}[/tex] then the triangle is an acute triangle.
2) Obtuse triangle: If the square of the longest side is greater than the sum of the square of the other two sides than the triangle is an obtuse triangle.
This implies that if, [tex](L_{3})^{2}>(L_{1})^{2}+(L_{2})^{2}[/tex] then the triangle is an obtuse triangle.
3) Right triangle:
If the square of the longest side is equal to the sum of the square of the other two sides than the triangle is a right triangle.
This implies that if, [tex](L_{3})^{2}=(L_{1})^{2}+(L_{2})^{2}[/tex] then the triangle is a right triangle.
Option1:
In option 1 it is given that the length of the sides of the triangle are [tex]L_{1}=4[/tex], [tex]L_{2}=5[/tex] and [tex]L_{3}=7[/tex].
The length of the longest side is [tex]7[/tex] units.
The square of the length of the longest side is calculated as follows:
[tex](L_{3})^{2}=49[/tex]
Calculate the value of [tex](L_{1})^{2}+(L_{2})^{2}[/tex] as follows:
[tex]\begin{aligned}(L_{1})^{2}+(L_{2})^{2}&=16+25\\&=41\end{aligned}[/tex]
This implies that for option 1 the square of the length of the longest side is greater than the sum of the squares of the other two sides. So, the triangle with sides [tex]L_{1}=4[/tex], [tex]L_{2}=5[/tex] and [tex]L_{3}=7[/tex] is an obtuse triangle.
Therefore, the option 1 is incorrect.
Option2:
In option 2 it is given that the length of the sides of the triangle are [tex]L_{1}=5[/tex], [tex]L_{2}=7[/tex] and [tex]L_{3}=8[/tex].
The length of the longest side is [tex]8[/tex] units.
The square of the length of the longest side is calculated as follows:
[tex](L_{3})^{2}=64[/tex]
Calculate the value of [tex](L_{1})^{2}+(L_{2})^{2}[/tex] as follows
[tex]\begin{aligned}(L_{1})^{2}+(L_{2})^{2}&=25+49\\&=74\end{aligned}[/tex]
This implies that for option 2 the square of the length of the longest side is smaller than the sum of the squares of the other two sides. So, the triangle with the side [tex]L_{1}=5[/tex], [tex]L_{2}=7[/tex] and [tex]L_{3}=8[/tex] is an acute triangle.
Therefore, the option 2 is correct.
Option3:
In option 3 it is given that the length of the sides of the triangle are [tex]L_{1}=6[/tex], [tex]L_{2}=7[/tex] and [tex]L_{3}=10[/tex].
The length of the longest side is [tex]10[/tex] units.
The square of the length of the longest side is calculated as follows:
[tex](L_{3})^{2}=100[/tex]
Calculate the value of [tex](L_{1})^{2}+(L_{2}){2}[/tex] as follows:
[tex]\begin{aligned}(L_{1})^{2}+(L_{2})^{2}&=36+49\\&=85\end{aligned}[/tex]
This implies that for option 3 the square of the length of the largest side is greater than the sum of the squares of the other two sides. So, the triangle with the side [tex]L_{1}=6[/tex], [tex]L_{2}=7[/tex] and [tex]L_{3}=10[/tex] is an obtuse triangle.
Therefore, the option 3 is incorrect.
Option4:
In option 4 it is given that the length of the sides of the triangle are [tex]L_{1}=7[/tex], [tex]L_{2}=9[/tex] and [tex]L_{3}=12[/tex].
The length of the longest side is [tex]12[/tex] units.
The square of the length of the longest side is calculated as follows:
[tex](L_{3})^{2}=144[/tex]
Calculate the value of [tex](L_{1})^{2}+(L_{2})^{2}[/tex] as follows:
[tex]\begin{aligned}(L_{1})^{2}+(L_{2})^{2}&=49+81\\&=130\end{aligned}[/tex]
This implies that for option 4 the square of the length of the largest side is greater than the sum of the squares of the other two sides. So, the triangle with the side [tex]L_{1}=7[/tex], [tex]L_{2}=9[/tex] and [tex]L_{3}=12[/tex] is an
obtuse triangle.
Therefore, the option 4 is incorrect.
Thus, the set of number which represent the length of the sides of an acute triangle is [tex]5,7,8[/tex] i.e., [tex]\fbox{\begin\\\ \bf option 2\\\end{minispace}}[/tex].
Learn more:
1. A problem to complete the square of quadratic function https://brainly.com/question/12992613
2. A problem to determine the slope intercept form of a line https://brainly.com/question/1473992
3. Inverse function https://brainly.com/question/1632445
Answer details
Grade: High school
Subject: Mathematics
Chapter: Triangles
Keywords: Geometry, triangles, acute triangle, obtuse triangles, right triangle, longest side, sum of longest side, classification of triangle, non-collinear points, 90 degrees, 5,7,8.
