Question

The coordinate values of the vertices of AKLM are integers.LKM5 4 3 219 1 2 3 4 5I12Which set of coordinate pairs could represent the vertices of a triangle congruent to AKLM?O {(0,0), (3, 4), (0,5)}O {(0,0).(-5,0), (0,4)}O {(-1, 1), (-4,5), (-1,5)}(11) (-14) 121)

Answer 1

Given the triangle KLM.

Let's find the set of coordinate pairs that represents the vertices of a triangle congruent to triangle KLM.

Congruent triangles have the same size. This means the corresponding side lengths of both triangles will be congruent.

We have the vertices of triangle KLM:

K(1, 1), L(5, 4), M(5, 1)

To find the coordinate pairs to a triangle from the list given, let's find the side length of triangle KLM using the distance formula:

`d=\sqrt[]{(x2-x1)^2+(y2-y1)^2_{}}`

Thus, we have:

Length of KL:

(x1, y1) ==> (1, 1)

(x2, y2) == (5, 4)

`\begin{gathered} KL=\sqrt[]{(5-1)^2+(4-1)^2}=\sqrt[]{4^2+3^2}=\sqrt[]{16+9}=\sqrt[]{25}=5 \\ \\ KL=5 \end{gathered}`

Length of KM:

(x1, y1) ==> (1, 1)

(x2, y2) ==> (5, 1)

`\begin{gathered} KM=\sqrt[]{(5-1)^2+(1-1)^2}=\sqrt[]{4^2+0^2}=4 \\ \\ KM=4 \end{gathered}`

Length of LM:

(x1, y1) ==> (5, 4)

(x2, y1) ==> (5, 1)

`\begin{gathered} LM=\sqrt[]{(5-5)^2+(1-4)^2}=\sqrt[]{0^2+3^2}=3 \\ \\ LM=3 \end{gathered}`

Therefore, the lengths of the side are:

KL = 5

KM = 4

LM = 3

From the choices given let's find the coordinate pairs that have the same distances with that of triangle KLM.

Let's check option C which have the coordinates:

{K'(-1, 1), L'(-4, 5), M'(-1, 5)]

Let's find the length of each side using the dstance formula.

Length of K'L':

(x1, y1) ==>(-1, 1)

(x2, y2) ==> (-4, 5)

`\begin{gathered} K^(\prime)L^(\prime)=\sqrt[]{(-4,-(-1))^2+(5-1)^2}=\sqrt[]{(-4+1)^2+(4)^2}=\sqrt[]{(-3)^2+4^2} \\ \\ K^(\prime)L^(\prime)=\sqrt[]{9+16}=\sqrt[]{25}=5 \\ \\ K^(\prime)L^(\prime)=5 \end{gathered}`

Length of K'M':

(x1, y1) ==> (-1, 1)

(x2, y2) ==> (-1, 5)

`\begin{gathered} K\text{'M'=}\sqrt[]{(-1-(-1))^2+(5-1)^2}=\sqrt[]{(-1+1)^2+(4)^2}=\sqrt[]{0+16}=4 \\ \\ K^(\prime)M^(\prime)=4 \end{gathered}`

Length of L'M':

(x1, y1) ==> (-4, 5)

(x2, y2) ==> (-1, 5)

`\begin{gathered} L^(\prime)M^(\prime)=\sqrt[]{(-1-(-4))^2+(5-5)^2}=\sqrt[]{(-1+4)^2+(0)^2}=\sqrt[]{3^2}=3 \\ \\ L^(\prime)M^(\prime)=5 \end{gathered}`

We have the side lengths:

K'M' = 5

K'M' = 4

L'M' = 3

We can see the coordinate pairs in option C have the same side lengths with the corresponding sides of the triangle KLM.

Since they have equal side lengths, we can say the triangle KLM and the triangle formed by the coordinate pairs in the third option are congruent triangles.

ANSWER:

{(-1, 1), (-4, 5), (-1, 5)}