Triangle Congruence using SSS and SAS | CK-12 Foundation - Free Printable

It looks like you've uploaded a coordinate plane with two triangles:

- Triangle DEF with vertices:
- D = (3, 9)
- E = (6, 4)
- F = (10, 7)

- Triangle ABC with vertices:
- A = (-2, -2)
- B = (-8, -6)
- C = (-6, -9)

Let’s analyze the problem. Since no specific question was stated, I’ll assume you want to determine whether these two triangles are congruent, similar, or related by some transformation (like translation, rotation, reflection, or dilation). Let's proceed step-by-step.

---

Step 1: Analyze the Position and Orientation

Looking at the coordinates:

- Triangle DEF is in the upper right quadrant.
- Triangle ABC is in the lower left quadrant.

This suggests that one triangle might be a transformation of the other.

Let’s check if there's a translation, rotation, reflection, or dilation that maps one triangle onto the other.

---

Step 2: Check for a Possible Transformation

Let’s suppose triangle ABC is a translated, rotated, or reflected version of triangle DEF. But first, let’s compare side lengths to see if they’re congruent or similar.

We will compute the side lengths using the distance formula:

$$
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

---

#### Compute side lengths of Triangle DEF:

1. DE: from D(3,9) to E(6,4)

$$
DE = \sqrt{(6 - 3)^2 + (4 - 9)^2} = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}
$$

2. EF: from E(6,4) to F(10,7)

$$
EF = \sqrt{(10 - 6)^2 + (7 - 4)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
$$

3. DF: from D(3,9) to F(10,7)

$$
DF = \sqrt{(10 - 3)^2 + (7 - 9)^2} = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53}
$$

So, sides of DEF: $ \sqrt{34}, 5, \sqrt{53} $

---

#### Compute side lengths of Triangle ABC:

1. AB: from A(-2,-2) to B(-8,-6)

$$
AB = \sqrt{(-8 + 2)^2 + (-6 + 2)^2} = \sqrt{(-6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52}
$$

2. BC: from B(-8,-6) to C(-6,-9)

$$
BC = \sqrt{(-6 + 8)^2 + (-9 + 6)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
$$

3. AC: from A(-2,-2) to C(-6,-9)

$$
AC = \sqrt{(-6 + 2)^2 + (-9 + 2)^2} = \sqrt{(-4)^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65}
$$

So, sides of ABC: $ \sqrt{52}, \sqrt{13}, \sqrt{65} $

Now compare:

- DEF: $ \sqrt{34}, 5, \sqrt{53} $
- ABC: $ \sqrt{52}, \sqrt{13}, \sqrt{65} $

These are not equal, so not congruent.

But maybe they are similar? Let’s check ratios.

But before that, let’s look for a possible transformation.

---

Step 3: Try to Find a Relationship Between Coordinates

Let’s try to see if there's a consistent vector or scaling between points.

Compare point D(3,9) and A(-2,-2)

Try to find a transformation that maps D → A:

- x: 3 → -2 → change of -5
- y: 9 → -2 → change of -11

Now check E(6,4) → B(-8,-6):

- x: 6 → -8 → change of -14
- y: 4 → -6 → change of -10 → not consistent with above

So not a simple translation.

Try to see if there's a rotation or reflection.

Alternatively, consider if one triangle is a scaled version of the other.

Let’s compare side lengths numerically:

| Side | DEF | ABC |
|------|-----|-----|
| DE | √34 ≈ 5.83 | AB = √52 ≈ 7.21 |
| EF | 5 | BC = √13 ≈ 3.61 |
| DF | √53 ≈ 7.28 | AC = √65 ≈ 8.06 |

Check ratios:

- AB / DE ≈ 7.21 / 5.83 ≈ 1.237
- AC / DF ≈ 8.06 / 7.28 ≈ 1.107 → not same ratio → not similar?

Wait — maybe we should match corresponding sides properly.

Let’s try matching by shape.

Look at triangle DEF:

- D(3,9), E(6,4), F(10,7)

Plotting mentally:
- From D to E: down and right
- From E to F: up and right
- From F to D: up and left

Triangle ABC:
- A(-2,-2), B(-8,-6), C(-6,-9)

From A to B: left and down
From B to C: right and down
From C to A: right and up

Not obviously symmetric.

Wait — perhaps the triangles are congruent via rotation or reflection.

Let’s try to find if there’s a central symmetry or 180° rotation.

Try checking if the midpoint between D and C is the same as between E and B, etc.

Wait — let’s try to see if the vector from D to E relates to vector from A to B, etc.

But perhaps a better idea: check if the triangles are congruent by measuring angles or looking for rigid motion.

Alternatively, let’s test if triangle ABC is a reflection or rotation of triangle DEF.

But here's an idea: try to see if the two triangles are reflections over the origin.

A point reflection over the origin means:
- (x, y) → (-x, -y)

Check:
- D(3,9) → (-3,-9) — but A is (-2,-2), not matching.
- E(6,4) → (-6,-4) — but B is (-8,-6), no.
- F(10,7) → (-10,-7) — not matching any point.

No.

What if it's a translation?

Try vector from D to A: (-2 - 3, -2 - 9) = (-5, -11)

Apply this to E(6,4): (6 - 5, 4 - 11) = (1, -7) → not B(-8,-6)

No.

Try vector from E to B: (-8 - 6, -6 - 4) = (-14, -10)

Apply to D(3,9): (3 - 14, 9 - 10) = (-11, -1) → not A

No.

Try scaling.

Suppose DEF is scaled and rotated to get ABC.

But let’s go back — maybe the two triangles are congruent but transformed.

Wait — let’s try computing vectors.

Another idea: maybe the triangles are reflections across the line y = x or y = -x, or some axis.

Or perhaps a 90° rotation.

Let’s try a 90° counterclockwise rotation about the origin:

Rule: (x, y) → (-y, x)

Apply to D(3,9): (-9, 3) — not near any point

90° clockwise: (x,y) → (y, -x)

D(3,9) → (9, -3) — not matching

Try 180°: (x,y) → (-x, -y)

D(3,9) → (-3,-9) — close to C(-6,-9)? Not really.

C is (-6,-9), which is not (-3,-9)

But wait — what if we try to see if triangle ABC is a rotation of triangle DEF by 180° around a point?

Let’s suppose there’s a center of rotation such that:

D → A
E → B
F → C

Then the center O must satisfy:
- O is midpoint of D and A
- O is midpoint of E and B
- O is midpoint of F and C

Check if midpoints are same.

Midpoint of D(3,9) and A(-2,-2):

$$
\left( \frac{3 + (-2)}{2}, \frac{9 + (-2)}{2} \right) = \left( \frac{1}{2}, \frac{7}{2} \right) = (0.5, 3.5)
$$

Midpoint of E(6,4) and B(-8,-6):

$$
\left( \frac{6 + (-8)}{2}, \frac{4 + (-6)}{2} \right) = \left( \frac{-2}{2}, \frac{-2}{2} \right) = (-1, -1)
$$

Not the same → not a 180° rotation around same point.

So not a rotation.

---

Alternative Idea: Are the Triangles Congruent?

Let’s compute all side lengths again and compare.

Triangle DEF:

- DE = √[(6-3)² + (4-9)²] = √[9 + 25] = √34 ≈ 5.83
- EF = √[(10-6)² + (7-4)²] = √[16 + 9] = √25 = 5
- DF = √[(10-3)² + (7-9)²] = √[49 + 4] = √53 ≈ 7.28

Triangle ABC:

- AB = √[(-8+2)² + (-6+2)²] = √[36 + 16] = √52 ≈ 7.21
- BC = √[(-6+8)² + (-9+6)²] = √[4 + 9] = √13 ≈ 3.61
- AC = √[(-6+2)² + (-9+2)²] = √[16 + 49] = √65 ≈ 8.06

Now sort both sets:

DEF: 5, √34≈5.83, √53≈7.28
ABC: √13≈3.61, √52≈7.21, √65≈8.06

No common side lengths → not congruent.

Also, ratios:

- √13 / 5 ≈ 0.72
- √52 / √34 ≈ 7.21 / 5.83 ≈ 1.237
- √65 / √53 ≈ 8.06 / 7.28 ≈ 1.107

Not proportional → not similar.

So neither congruent nor similar.

But wait — maybe I made a mistake in matching the points.

Perhaps the corresponding vertices are different.

Maybe D corresponds to C, E to B, F to A?

Let’s try pairing differently.

Suppose:

- D(3,9) ↔ C(-6,-9)
- E(6,4) ↔ B(-8,-6)
- F(10,7) ↔ A(-2,-2)

Check distances:

- DC = √[(-6-3)² + (-9-9)²] = √[(-9)² + (-18)²] = √[81 + 324] = √405 = 9√5 ≈ 20.12 — too big

No.

Alternatively, maybe the triangles are related by a dilation and rotation.

But let’s take a different approach.

---

Look at the Graph Carefully

Notice:

- Triangle DEF has points in upper right
- Triangle ABC has points in lower left

But more importantly — are the triangles mirror images or rotated versions?

Let’s try plotting the vector differences.

Wait — here's a key insight:

Let’s check if triangle ABC is a 180° rotation of triangle DEF around the origin?

Try:

- D(3,9) → (-3,-9) — but no point at (-3,-9)
- E(6,4) → (-6,-4)
- F(10,7) → (-10,-7)

But our ABC points are:
- A(-2,-2)
- B(-8,-6)
- C(-6,-9)

None match.

But wait — what if we try to see if there's a translation that maps DEF to ABC?

Suppose we shift DEF by vector (dx, dy) to get ABC.

Assume D → A: (3,9) → (-2,-2) → dx = -5, dy = -11

Apply to E(6,4): (6-5, 4-11) = (1, -7) → not B(-8,-6)

No.

Try D → B: (3,9) → (-8,-6): dx = -11, dy = -15

Then E(6,4) → (6-11, 4-15) = (-5,-11) — not A or C

No.

Try D → C: (3,9) → (-6,-9): dx = -9, dy = -18

Then E(6,4) → (6-9, 4-18) = (-3,-14) — not B or A

No.

So no translation.

---

Final Idea: Perhaps the triangles are congruent via reflection or rotation, but let's calculate the area.

Compute area using shoelace formula.

#### Area of Triangle DEF:

Points: D(3,9), E(6,4), F(10,7)

Shoelace:

$$
\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|
$$

Plug in:

= ½ |3(4 - 7) + 6(7 - 9) + 10(9 - 4)|
= ½ |3(-3) + 6(-2) + 10(5)|
= ½ |-9 -12 + 50| = ½ |29| = 14.5

#### Area of Triangle ABC:

A(-2,-2), B(-8,-6), C(-6,-9)

= ½ |(-2)(-6 - (-9)) + (-8)(-9 - (-2)) + (-6)(-2 - (-6))|

= ½ |(-2)(3) + (-8)(-7) + (-6)(4)|
= ½ |-6 + 56 - 24| = ½ |26| = 13

Areas: DEF = 14.5, ABC = 13 → not equal → not congruent.

So not congruent.

But wait — could there be a typo or mislabeling?

Wait — let’s double-check the shoelace for ABC.

Points: A(-2,-2), B(-8,-6), C(-6,-9)

List in order: A, B, C, back to A

Shoelace:

x: -2, -8, -6, -2
y: -2, -6, -9, -2

Sum1 = (-2)(-6) + (-8)(-9) + (-6)(-2) = 12 + 72 + 12 = 96

Sum2 = (-2)(-8) + (-6)(-6) + (-9)(-2) = 16 + 36 + 18 = 70

Area = ½ |96 - 70| = ½ × 26 = 13 ✔

DEF: 14.5

So areas are different.

Thus, not congruent, not similar (since side ratios aren't constant), no obvious transformation.

But wait — perhaps the task is to find the transformation that maps one to the other.

Alternatively, maybe the problem is to determine if they are congruent or similar, or to find the image under a transformation.

Since no explicit question was given, I suspect the intended task is to identify the relationship.

But let’s try one last thing: look at the vectors.

Let’s compute vectors:

For DEF:

- Vector DE = E - D = (6-3, 4-9) = (3, -5)
- Vector DF = F - D = (10-3, 7-9) = (7, -2)

For ABC:

- Vector AB = B - A = (-8+2, -6+2) = (-6, -4)
- Vector AC = C - A = (-6+2, -9+2) = (-4, -7)

Now, compare:

Is there a linear transformation (e.g., rotation + scaling) that maps (3,-5) to (-6,-4)?

Suppose scale factor k.

If (3,-5) → (-6,-4), then k = -2 for x, but -4/-5 = 0.8 — not same.

Not a uniform scaling.

But notice:

- DE = (3,-5)
- AB = (-6,-4) — not parallel

Wait — what if we try to see if the triangles are reflections?

Try reflecting over the origin: (x,y) → (-x,-y)

Then D(3,9) → (-3,-9) — not a vertex

But C is (-6,-9), B is (-8,-6), A is (-2,-2)

Closest is (-3,-9) — not there.

But wait — look at the coordinates:

- D(3,9), C(-6,-9) — x: 3 → -6, y: 9 → -9 → x multiplied by -2, y by -1 — not consistent

- E(6,4), B(-8,-6): 6→-8, 4→-6 — not proportional

- F(10,7), A(-2,-2): 10→-2, 7→-2 — no

But here's a possibility: maybe the two triangles are not related by a single transformation, but the problem is to find something else.

Wait — perhaps the task is to name the type of transformation or find the coordinates after a transformation.

But since no instruction was given, and based on the way the image is presented, it's likely that the intended question is:

> "Are the two triangles congruent? If so, what transformation maps one to the other?"

Based on our analysis:

- Areas are different: 14.5 vs 13
- Side lengths don't match
- No consistent transformation

Therefore, the two triangles are not congruent, and not similar.

But wait — let’s check if they are congruent by a different correspondence.

Try pairing:

- D(3,9) ↔ B(-8,-6)
- E(6,4) ↔ C(-6,-9)
- F(10,7) ↔ A(-2,-2)

Compute distances:

- DB = √[(-8-3)² + (-6-9)²] = √[(-11)² + (-15)²] = √[121 + 225] = √346

Too large.

Alternatively, perhaps the problem is to find the length of a side or area.

But without a specific question, it's hard to know.

---

Conclusion:

Given the lack of a specific question, but based on typical problems of this type, the most likely task is to determine if the two triangles are congruent or similar.

After detailed analysis:

- Side lengths are not equal
- Areas are different (14.5 vs 13)
- No consistent transformation (translation, rotation, reflection, dilation) maps one to the other
- Ratios of sides are not proportional

✔ Final Answer:

The two triangles are neither congruent nor similar. There is no rigid transformation or dilation that maps one triangle onto the other.

If you have a specific question (e.g., "Find the area of triangle DEF", or "What transformation maps DEF to ABC?", please provide it for a more targeted solution.