Calculate Triangle ABC Area: Using 12-Unit Rectangle and 13-Unit Diagonal

Question

Look at the following rectangle:

Calculate the area of the triangle ABC.

00:04 Let's calculate the area of triangle ABC.

00:09 First, we will use the Pythagorean theorem in triangle ABC to find side BC.

00:15 We will substitute the given values, and calculate to find the length of BC.

00:36 Now, let's isolate BC to find its exact value.

00:56 This is the length of BC.

01:00 Now, we'll use the formula for the area of a triangle.

01:05 That's height times the base, divided by 2.

01:09 We'll input the appropriate values, and calculate to find the area.

01:22 And that's how we solve the problem.

Let's solve this step-by-step:

Step 1: Identify the given information.
We know the rectangle $ABCD$ , is divided by its diagonal $AC$ . The length $AB$ is $12$ , and the diagonal $AC$ is $13$ .
Step 2: Apply Pythagorean theorem to find $BC$ , which acts as the height.
Using the Pythagorean theorem in $\triangle ABC$ gives us: $AC = \sqrt{AB^2 + BC^2}$ Given $AC = 13$ and $AB = 12$ , we set up the equation: $13 = \sqrt{12^2 + BC^2}$ Squaring both sides leads to: $169 = 144 + BC^2$ $BC^2 = 169 - 144 = 25$ Thus, $BC = \sqrt{25} = 5$ .
Step 3: Calculate the area of $\triangle ABC$ .
The area can be found using the formula: $\text{Area of } \triangle ABC = \frac{1}{2} \times AB \times BC$ $= \frac{1}{2} \times 12 \times 5$ $= \frac{1}{2} \times 60 = 30$

Therefore, the area of triangle $ABC$ is $\boxed{30}$ .