Verified Answer :
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 \(5,7,8\) i.e., \(\fbox{\begin\\\ \bf option 2\\\end{minispace}}\).
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 \(L_{1},L_{2} \text{and} L_{3}\) such that \(L_{3}>L_{2}>L_{1}\).
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, \((L_{3})^{2}<(L_{1})^{2}+(L_{2})^{2}\) 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, \((L_{3})^{2}>(L_{1})^{2}+(L_{2})^{2}\) 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, \((L_{3})^{2}=(L_{1})^{2}+(L_{2})^{2}\) then the triangle is a right triangle.
Option1:
In option 1 it is given that the length of the sides of the triangle are \(L_{1}=4\), \(L_{2}=5\) and \(L_{3}=7\).
The length of the longest side is \(7\) units.
The square of the length of the longest side is calculated as follows:
\((L_{3})^{2}=49\)
Calculate the value of \((L_{1})^{2}+(L_{2})^{2}\) as follows:
\(\begin{aligned}(L_{1})^{2}+(L_{2})^{2}&=16+25\\&=41\end{aligned}\)
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 \(L_{1}=4\), \(L_{2}=5\) and \(L_{3}=7\) 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 \(L_{1}=5\), \(L_{2}=7\) and \(L_{3}=8\).
The length of the longest side is \(8\) units.
The square of the length of the longest side is calculated as follows:
\((L_{3})^{2}=64\)
Calculate the value of \((L_{1})^{2}+(L_{2})^{2}\) as follows
\(\begin{aligned}(L_{1})^{2}+(L_{2})^{2}&=25+49\\&=74\end{aligned}\)
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 \(L_{1}=5\), \(L_{2}=7\) and \(L_{3}=8\) 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 \(L_{1}=6\), \(L_{2}=7\) and \(L_{3}=10\).
The length of the longest side is \(10\) units.
The square of the length of the longest side is calculated as follows:
\((L_{3})^{2}=100\)
Calculate the value of \((L_{1})^{2}+(L_{2}){2}\) as follows:
\(\begin{aligned}(L_{1})^{2}+(L_{2})^{2}&=36+49\\&=85\end{aligned}\)
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 \(L_{1}=6\), \(L_{2}=7\) and \(L_{3}=10\) 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 \(L_{1}=7\), \(L_{2}=9\) and \(L_{3}=12\).
The length of the longest side is \(12\) units.
The square of the length of the longest side is calculated as follows:
\((L_{3})^{2}=144\)
Calculate the value of \((L_{1})^{2}+(L_{2})^{2}\) as follows:
\(\begin{aligned}(L_{1})^{2}+(L_{2})^{2}&=49+81\\&=130\end{aligned}\)
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 \(L_{1}=7\), \(L_{2}=9\) and \(L_{3}=12\) 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 \(5,7,8\) i.e., \(\fbox{\begin\\\ \bf option 2\\\end{minispace}}\).
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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.