Question

are these two triangles similar? if so, write the scale factor.
are these two triangles similar? yes or no?
if so, write the scale factor. if not write none

Explanation:

Step1: Determine coordinates of triangle ABC

Point A: (-6, 5), Point B: (6, 0), Point C: (-6, 0). So, AC is vertical with length \( 5 - 0 = 5 \), BC is horizontal with length \( 6 - (-6) = 12 \).

Step2: Determine coordinates of triangle PQR

Point Q: (-4, -2), Point R: (3, -2), Point P: (3, -5). So, PR is vertical with length \( -2 - (-5) = 3 \), QR is horizontal with length \( 3 - (-4) = 7 \). Wait, correction: Wait, QR is from (-4,-2) to (3,-2), length \( 3 - (-4) = 7 \)? No, wait, original triangle ABC: AC is from (-6,5) to (-6,0), length 5. BC is from (-6,0) to (6,0), length 12. Triangle PQR: PR is from (3,-2) to (3,-5), length \( |-2 - (-5)| = 3 \). QR is from (-4,-2) to (3,-2), length \( |3 - (-4)| = 7 \)? Wait, no, maybe I mixed up the triangles. Wait, triangle ABC: right triangle with legs AC (vertical) and BC (horizontal). Triangle PQR: right triangle with legs PR (vertical) and QR (horizontal). Wait, let's recalculate lengths correctly.

For triangle ABC:
- AC: vertical distance from A(-6,5) to C(-6,0): \( 5 - 0 = 5 \)
- BC: horizontal distance from C(-6,0) to B(6,0): \( 6 - (-6) = 12 \)

For triangle PQR:
- PR: vertical distance from R(3,-2) to P(3,-5): \( |-2 - (-5)| = 3 \)
- QR: horizontal distance from Q(-4,-2) to R(3,-2): \( |3 - (-4)| = 7 \)

Wait, that can't be. Wait, maybe the red triangle is QRP? Wait, no, the red triangle has points Q(-4,-2), R(3,-2), P(3,-5). So it's a right triangle at R. So legs: QR (horizontal) from Q to R: length 7, PR (vertical) from R to P: length 3.

Triangle ABC: right triangle at C. Legs: AC (vertical) length 5, BC (horizontal) length 12.

Wait, maybe I made a mistake. Let's check the slopes. The hypotenuse of ABC: from A(-6,5) to B(6,0). Slope: \( \frac{0 - 5}{6 - (-6)} = \frac{-5}{12} \).

Hypotenuse of PQR: from Q(-4,-2) to P(3,-5). Slope: \( \frac{-5 - (-2)}{3 - (-4)} = \frac{-3}{7} \). Wait, that's not the same. Wait, maybe the triangles are ABC and another triangle. Wait, maybe the red triangle is QPR? No, the coordinates: Q(-4,-2), P(3,-5), R(3,-2). Wait, maybe I flipped the legs. Wait, maybe triangle ABC: AC is 5, BC is 12. Triangle PQR: PR is 3, QR is 7? No, that can't be. Wait, maybe the grid is 1 unit per square. Let's count the units again.

Triangle ABC:
- A(-6,5), C(-6,0): 5 units up (since y from 0 to 5, x same).
- C(-6,0), B(6,0): 12 units right (x from -6 to 6, y same).

Triangle PQR:
- R(3,-2), P(3,-5): 3 units down (y from -2 to -5, x same).
- Q(-4,-2), R(3,-2): 7 units right (x from -4 to 3, y same). Wait, that's 7? No, -4 to 3 is 7 units? 3 - (-4) = 7. Yes.

Wait, but 5/3 = 5/3, 12/7 ≈ 1.714, not equal. Wait, maybe I messed up the triangles. Wait, maybe the red triangle is Q(-4,-2), R(3,-2), P(3,-5) – no, maybe the other triangle. Wait, maybe the problem is that triangle ABC and triangle PQR are similar? Wait, no, maybe I made a mistake in coordinates. Wait, let's check the y-axis. The top triangle: A is at (-6,5), C at (-6,0), B at (6,0). So AC length: 5 (from y=0 to y=5). BC length: 12 (from x=-6 to x=6). The bottom triangle: Q at (-4,-2), R at (3,-2), P at (3,-5). So PR length: 3 (from y=-2 to y=-5). QR length: 7 (from x=-4 to x=3). Wait, that's not matching. Wait, maybe the scale factor is 5/3? Wait, no, 5 and 3: 5/3, 12 and 7: no. Wait, maybe I flipped the legs. Wait, triangle ABC: AC=5, BC=12. Triangle PQR: PR=3, QR=7. No, that's not proportional. Wait, maybe the triangles are ABC and another triangle. Wait, maybe the red triangle is Q(-4,-2), R(3,-2), P(3,-5) – no, maybe the coordinates are different. Wait, maybe the grid is 1 unit, but let's check the lengths again.

Wait, maybe I made a mistake in the vertical leg of triangle ABC. A is at (-6,5), C at (-6,0): so vertical length is 5 (from y=0 to y=5). Correct. BC: from (-6,0) to (6,0): horizontal length 12. Correct.

Triangle PQR: R is at (3,-2), P at (3,-5): vertical length 3 (from y=-2 to y=-5). Q at (-4,-2), R at (3,-2): horizontal length 7 (from x=-4 to x=3). Wait, 5/3 = 5/3, 12/7 ≈ 1.714, not equal. So they are not similar? But that can't be. Wait, maybe the red triangle is Q(-4,-2), P(3,-5), and R(3,-2) – no, maybe the other triangle. Wait, maybe the problem is that the triangles are similar with scale factor 5/3? Wait, no, 5 and 3: 5/3, 12 and 7: no. Wait, maybe I messed up the horizontal leg of triangle PQR. Wait, Q is at (-4,-2), R at (3,-2): x from -4 to 3 is 7 units? Yes. R at (3,-2), P at (3,-5): y from -2 to -5 is 3 units. So legs 7 and 3. Triangle ABC: legs 5 and 12. 5/3 = 5/3, 12/7 ≈ 1.714, not equal. So they are not similar? But that seems odd. Wait, maybe the triangles are ABC and another triangle. Wait, maybe the red triangle is Q(-4,-2), R(3,-2), P(3,-5) – no, maybe the coordinates are different. Wait, maybe the vertical leg of triangle ABC is 5, and the vertical leg of triangle PQR is 3, and horizontal legs 12 and 7? No, that's not proportional. So the answer would be no, scale factor none. But wait, maybe I made a mistake. Wait, let's check the slopes again. Slope of AB: (0-5)/(6 - (-6)) = -5/12. Slope of QP: (-5 - (-2))/(3 - (-4)) = -3/7. Not equal, so angles are different. So triangles are not similar.

Answer:

Are these two triangles similar? no
If so, write the scale factor. If not write 'none': none