Question

Which set of numbers can represent the side lengths, in inches, of an acute triangle? 4, 5, 7 5, 7, 8 6, 7, 10 7, 9, 12

Answer 1

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.

Answer 2

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, $(L_{3})^{2}$ 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., $\fbox{\begin\\\ \bf option 2\\\end{minispace}}$.

Learn more:  

1. A problem to complete the square of quadratic function brainly.com/question/12992613  

2. A problem to determine the slope intercept form of a line brainly.com/question/1473992

3. Inverse function 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.