APPLY THE RULE [1.5X, 1.5Y] TO THE VERTICES OF TRIANGLE TO GET TRINGLE PQR. COMPARE THE CORRESPONDING MEASUREMENTS[SIDE LENGTHS, PERIMETER, AREA, ANGLE MEASURES] OF THE TRIANGLE
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To apply the rule [1.5X, 1.5Y] to the vertices of a triangle, we need to multiply the coordinates of each vertex by their corresponding scaling factors.
Let's say the original triangle ABC has vertices with coordinates A(Xa, Ya), B(Xb, Yb), and C(Xc, Yc). Applying the scaling factors, the coordinates of the new triangle PQR would be:
P(Xp, Yp) = [1.5 * Xa, 1.5 * Ya]
Q(Xq, Yq) = [1.5 * Xb, 1.5 * Yb]
R(Xr, Yr) = [1.5 * Xc, 1.5 * Yc]
Now, let's compare the corresponding measurements of the two triangles:
1. Side lengths: Calculate the lengths of the sides of triangles ABC and PQR using the distance formula:
Side AB = √((Xb - Xa)^2 + (Yb - Ya)^2)
Side PQ = √((Xq - Xp)^2 + (Yq - Yp)^2)
Similarly, calculate the lengths of side BC and side AC, as well as side QR and side PR.
2. Perimeter: Add up the lengths of the three sides of each triangle to get the perimeter.
Perimeter of ABC = Side AB + Side BC + Side AC
Perimeter of PQR = Side PQ + Side QR + Side PR
3. Area: Calculate the area of each triangle using the Shoelace formula or Heron's formula.
Area of ABC = 0.5 * |(Xa * Yb + Xb * Yc + Xc * Ya) - (Xb * Ya + Xc * Yb + Xa * Yc)|
Area of PQR = 0.5 * |(Xp * Yq + Xq * Yr + Xr * Yp) - (Xq * Yp + Xr * Yq + Xp * Yr)|
4. Angle measures: The angle measures of triangles ABC and PQR would not change because angles are determined by the relationship between the sides and not the specific coordinates of the vertices.
Compare the side lengths, perimeter, area, and angle measures of the two triangles to determine the differences resulting from the scaling rule [1.5X, 1.5Y].
To apply the rule [1.5X, 1.5Y] to the vertices of a triangle, you need to multiply the coordinates of each vertex by the scaling factors 1.5X and 1.5Y. This will result in the new coordinates of the vertices of the transformed triangle PQR.
Let's assume the original triangle has vertices A, B, and C. The coordinates of these vertices are (Ax, Ay), (Bx, By), and (Cx, Cy) respectively.
To apply the rule [1.5X, 1.5Y], you need to multiply each coordinate by the corresponding scaling factor. Therefore, the new coordinates of the transformed triangle PQR will be:
Px = 1.5X * Ax
Py = 1.5Y * Ay
Qx = 1.5X * Bx
Qy = 1.5Y * By
Rx = 1.5X * Cx
Ry = 1.5Y * Cy
Once you have the coordinates of the transformed triangle PQR, you can compare various measurements with the original triangle ABC:
1. Side Lengths:
Calculate the length of each side of both triangles using the distance formula:
- For the original triangle ABC, calculate the lengths of AB, BC, and AC.
- For the transformed triangle PQR, calculate the lengths of PQ, QR, and RP.
Compare the lengths of corresponding sides to see if there are any changes.
2. Perimeter:
Add up the lengths of all three sides for both triangles.
- For the original triangle ABC, calculate the perimeter by adding AB + BC + AC.
- For the transformed triangle PQR, calculate the perimeter by adding PQ + QR + RP.
Compare the perimeters of both triangles.
3. Area:
Calculate the area of each triangle using any appropriate area formula (such as Heron's formula or the formula for the area of a triangle given its base and height).
- For the original triangle ABC, calculate the area.
- For the transformed triangle PQR, calculate the area.
Compare the areas of both triangles.
4. Angle Measures:
Calculate the measures of each angle in both triangles using trigonometry, such as the Law of Cosines or Law of Sines.
- For the original triangle ABC, calculate the angles at A, B, and C.
- For the transformed triangle PQR, calculate the angles at P, Q, and R.
Compare the angle measures of corresponding angles.
By comparing all these measurements, you can determine how the rule [1.5X, 1.5Y] affects the side lengths, perimeter, area, and angle measures of the triangle.