Create an image of two triangles positioned on a blank Cartesian plane. The first triangle, labeled as ABC, has its vertices at A(3, 1), B(4, 5), and C(2, 3). The second triangle, named DEF, has its vertices located at D(-1, -3), E(-5, -4), and F(-3, -2). The triangles should be geometrically congruent to each other. Please ensure that the image is presented without any additional text.

Prove that the triangles with the given vertices are congruent.

A(3, 1), B(4, 5), C(2, 3)

D(-1, -3), E(-5, -4), F(-3, -2)

a
The triangle are congruent because triangle ABC can be mapped to triangle DEF by a rotation. (x,y)--->(y, -x) Followed by a reflection (x, y)--->(x, -y)
b
The triangle are congruent because triangle ABC can be mapped to triangle DEF by a reflection. (x, y)---> (-x, y) Followed by a rotation (x, y)---> (-x, y)
c
The triangle are congruent because triangle ABC can be mapped to triangle DEF by a translation. (x, y)--->(x-4, y) Followed by translation (x,y)---> (x, y-6)
d
The triangle are congruent because triangle ABC can be mapped to triangle DEF by a rotation. (x,y)---> (-y, x) Followed by a reflection (x, y)--->(x, -y)

is it b?

Is it d?

Rectangle QRST has vertices Q(-6, -1), R(-3, 1), S(1,-5),andT(-2,-7) with the translation rule: (x,y)→(x+5,y+7). Q' would be

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No, it is not option b. Option b suggests that the triangles can be mapped to each other by a reflection followed by a rotation, but this is not true. The correct answer is option a. The triangles can be mapped to each other by a rotation (x, y) ⟶ (y, -x) followed by a reflection (x, y) ⟶ (x, -y).

No, it is not option b. To prove that two triangles are congruent, we need to show that all corresponding sides and angles are equal.

To determine if the triangles with vertices A(3, 1), B(4, 5), C(2, 3) and D(-1, -3), E(-5, -4), F(-3, -2) are congruent, we can use the distance formula to find the lengths of the sides and compare them.

The distance between two points (x1, y1) and (x2, y2) is given by the formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

So let's calculate the distances of the sides for both triangles.

For triangle ABC:
Side AB: d(AB) = sqrt((4 - 3)^2 + (5 - 1)^2) = sqrt(1^2 + 4^2) = sqrt(1 + 16) = sqrt(17)
Side BC: d(BC) = sqrt((2 - 4)^2 + (3 - 5)^2) = sqrt((-2)^2 + (-2)^2) = sqrt(4 + 4) = sqrt(8) = 2sqrt(2)
Side AC: d(AC) = sqrt((2 - 3)^2 + (3 - 1)^2) = sqrt((-1)^2 + 2^2) = sqrt(1 + 4) = sqrt(5)

For triangle DEF:
Side DE: d(DE) = sqrt((-5 - (-1))^2 + (-4 - (-3))^2) = sqrt((-5 + 1)^2 + (-4 + 3)^2) = sqrt((-4)^2 + (-1)^2) = sqrt(16 + 1) = sqrt(17)
Side EF: d(EF) = sqrt((-3 - (-5))^2 + (-2 - (-4))^2) = sqrt((-3 + 5)^2 + (-2 + 4)^2) = sqrt((2)^2 + (2)^2) = sqrt(4 + 4) = sqrt(8) = 2sqrt(2)
Side DF: d(DF) = sqrt((-3 - (-1))^2 + (-2 - (-3))^2) = sqrt((-3 + 1)^2 + (-2 + 3)^2) = sqrt((-2)^2 + 1^2) = sqrt(4 + 1) = sqrt(5)

By comparing the side lengths, we can see that d(AB) = d(DE), d(BC) = d(EF), and d(AC) = d(DF), which indicates that the corresponding sides of the triangles are equal. Therefore, the triangles with the given vertices are congruent.

So, the correct answer is not b; it is a.

Nope. B just undoes itself.

Look carefully at the numbers and check the others.