Question

What happens to the area of a right triangle if each side is doubled in length? A. It increases by a factor of 1.41 . B. It decreases by a factor of 1.41 . C. It increases by a factor of 4 . D. It decreases by a factor of 4 .

Short answer

C. It increases by a factor of 4.

Step-by-step solution

Identify the Properties of a Right Triangle

A right triangle has two legs perpendicular to each other and a hypotenuse. The area of a right triangle is given by the formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)

Determine the New Side Lengths

If each side of the right triangle is doubled, then the base and height become twice their original lengths. Let the original base be \( b \) and the original height be \( h \). The new base will be \( 2b \) and the new height will be \( 2h \).

Calculate the New Area

Substitute the new base and height into the area formula: \( \text{New Area} = \frac{1}{2} \times 2b \times 2h \). Simplifying this gives: \( \text{New Area} = \frac{1}{2} \times 4bh = 2bh \).

Compare the New Area to the Original Area

The original area is \( \frac{1}{2} \times b \times h \). Compare this to the new area: \( \text{New Area} = 4 \times \text{Original Area} \). Therefore, the area increases by a factor of 4.

Key concepts

Right Triangle

A right triangle is a type of triangle that has one angle equal to 90 degrees. This right angle is formed where the two legs meet.
These legs are the sides of the triangle that are perpendicular to each other, often referred to as the base and the height.
The side opposite the right angle is called the hypotenuse, which is the longest side of the right triangle.

For example, in a right triangle with legs of lengths 3 and 4, the hypotenuse would have a length of 5, according to the Pythagorean theorem: \(3^2 + 4^2 = 5^2\).
Remembering these properties is crucial because it helps us in calculations involving right triangles.

Area Calculation

Calculating the area of a right triangle is straightforward. The area of a right triangle is defined by the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
This formula is derived from the general triangle area formula, where one half of the product of the base and height gives the area.

Let's say you have a right triangle with a base of 5 units and a height of 6 units.
Using the formula, the area would be: \( \text{Area} = \frac{1}{2} \times 5 \times 6 = 15 \) square units.
Understanding this formula is key when considering changes to the triangle, such as what happens to its area if we change the length of its sides.

Scaling Effects

When we scale a geometric shape, such as a right triangle, by increasing the lengths of its sides, certain effects come into play.
In this exercise, we look specifically at what happens when each side of a right triangle is doubled.
If the original base and height of the right triangle are \(b\) and \(h\) respectively, doubling these would give new dimensions 2\text{b} and 2h.

Using the area formula with these new dimensions: \(\text{New Area} = \frac{1}{2} \times 2b \times 2h = 2 \times bh \).
This shows that the area of the right triangle has increased by a factor of 4, because \(2bh\) is four times the original \(\frac{1}{2} \times b \times h = \frac{bh}{2} \).
This demonstrates an important principle of scaling: if the lengths of a shape are scaled by a factor, the area is scaled by the square of that factor.