Question

tinh has a triangle with side lengths 12 mm, 15 mm, and 18 mm. he wants to reduce the triangle by one-third, so he divides each side length and each angle measure by 3. which describes tinhs reduction?

tinhs reduction is incorrect; he should have only divided the side lengths by 3.
tinhs reduction is incorrect; he should have only divided the angle measures by 3.
tinhs reduction is incorrect; he should have multiplied the side lengths by 3 instead.
tinhs reduction is incorrect; he should have multiplied the angle measures by 3 instead.

Explanation:

Analyze the geometric reduction process

Tinh wants to reduce a triangle by a scale factor of \(\frac{1}{3}\). Under a geometric dilation (reduction or enlargement), the side lengths of the figure are scaled proportionally, but the angle measures must remain constant to preserve the shape of the figure.

Evaluate the effect on angle measures

Using the Triangle Angle Sum Theorem, we know that the sum of the interior angles of any triangle is always exactly \(180^\circ\). If Tinh divides each angle measure by \(3\), the sum of the new angles would be:

$$ \frac{\text{Sum of original angles}}{3} = \frac{180^\circ}{3} = 60^\circ $$

A triangle cannot have an interior angle sum of \(60^\circ\). Therefore, dividing the angle measures is mathematically impossible and incorrect.

Determine the correct reduction method

To correctly perform a reduction, only the side lengths should be divided by \(3\) (multiplied by \(\frac{1}{3}\)). The corresponding angles of the reduced triangle must remain congruent to the original angles to maintain similarity. Thus, Tinh's reduction is incorrect because he should have only divided the side lengths by \(3\).

Answer:

• (A) Tinh's reduction is incorrect; he should have only divided the side lengths by 3. (Correct answer)
• (B) Tinh's reduction is incorrect; he should have only divided the angle measures by 3.
• (C) Tinh's reduction is incorrect; he should have multiplied the side lengths by 3 instead.
• (D) Tinh's reduction is incorrect; he should have multiplied the angle measures by 3 instead.