Question

triangle j is shown below. james drew a scaled version of triangle j using a scale factor of 4 and labeled it triangle k. what is the area of triangle k? square units. (there is an image of triangle j with sides labeled 4 and 5)

Explanation:

Step1: Determine if Triangle J is a right triangle

We can check if Triangle J is a right triangle using the Pythagorean theorem. Let's assume the sides are \(a = 3\) (we can find the third side using the Pythagorean triple, since \(3 - 4 - 5\) is a Pythagorean triple), \(b = 4\), and \(c = 5\). So, \(3^2+4^2 = 9 + 16=25 = 5^2\), so Triangle J is a right triangle with legs 3 and 4.

Step2: Calculate the area of Triangle J

The area of a right triangle is given by \(A=\frac{1}{2}\times\text{base}\times\text{height}\). For Triangle J, base \(b = 4\) and height \(h = 3\) (or vice - versa). So, \(A_J=\frac{1}{2}\times3\times4 = 6\) square units.

Step3: Determine the scale factor effect on area

When a figure is scaled by a scale factor \(k\), the area of the scaled figure is \(k^{2}\) times the area of the original figure. Here, the scale factor \(k = 4\).

Step4: Calculate the area of Triangle K

The area of Triangle K, \(A_K=k^{2}\times A_J\). Substituting \(k = 4\) and \(A_J = 6\), we get \(A_K=4^{2}\times6=16\times6 = 96\) square units.

Answer:

96