Math  /  Geometry

QuestionPart 1: What is the SCALE FACTOR going from big to small? (leave it as a fraction or round the decimal to the nearest thousandths)? \square Part 2: What is the PERIMETER of the smaller figure? \square

Studdy Solution

STEP 1

1. The two triangles are similar.
2. Similar triangles have proportional sides.
3. The scale factor is the ratio of corresponding sides from the larger triangle to the smaller triangle.
4. The perimeter of a triangle is the sum of its side lengths.

STEP 2

1. Determine the scale factor from the larger triangle to the smaller triangle.
2. Use the scale factor to find the missing side of the smaller triangle.
3. Calculate the perimeter of the smaller triangle.

STEP 3

To find the scale factor, we need to compare corresponding sides of the triangles. We can use the side labeled 42 42 in the larger triangle and 25 25 in the smaller triangle. The scale factor k k from the larger triangle to the smaller triangle is:
k=side of smaller trianglecorresponding side of larger triangle k = \frac{\text{side of smaller triangle}}{\text{corresponding side of larger triangle}} k=2542 k = \frac{25}{42}

STEP 4

Now, we use the scale factor to find the missing side of the smaller triangle. We can use the side labeled 48 48 in the larger triangle. Let x x be the missing side of the smaller triangle. Then:
x48=2542 \frac{x}{48} = \frac{25}{42}

STEP 5

Solve for x x by cross-multiplying:
x=2542×48 x = \frac{25}{42} \times 48

STEP 6

Calculate the value of x x :
x=25×4842 x = \frac{25 \times 48}{42} x=120042 x = \frac{1200}{42} x28.571 x \approx 28.571

STEP 7

Now, calculate the perimeter of the smaller triangle. The sides are 35 35 , 25 25 , and 28.571 28.571 . The perimeter P P is:
P=35+25+28.571 P = 35 + 25 + 28.571

STEP 8

Calculate the perimeter:
P=35+25+28.571 P = 35 + 25 + 28.571 P=88.571 P = 88.571
The scale factor from the larger triangle to the smaller triangle is:
2542 \boxed{\frac{25}{42}}
The perimeter of the smaller triangle is:
88.571 \boxed{88.571}

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