Diagonals: Calculating area

To solve this problem, we'll follow these steps:

• Step 1: Use the Pythagorean Theorem to calculate the length of $AD$.
• Step 2: Calculate the area of triangle $\triangle BCD$.

Step 1: Given $AB = 12$ and $AC = 13$, we use the Pythagorean Theorem to find $AD$.

Step 2: Knowing the sides $AD = 5$ (height of the rectangle) and $AB = 12$ (base of the rectangle), triangle $\triangle BCD$ will have the base $BC = 12$ and the height $BD = 5$.

The area of triangle $\triangle BCD$ is:

Therefore, the area of triangle $\triangle BCD$ is 30.

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 $\boxed{30}$.

According to the given information, we can claim that:

$BD=2BO=8.5\times2=17$

Now let's look at triangle ABD to calculate side AB

$AB^2+AD^2=BD^2$

Let's input the known data:

$AB^2+8^2=17^2$

$AB^2=289-64=225$

We'll take the square root

$AB=15$

Now let's calculate the area of triangle ABD:

$\frac{15\times8}{2}=\frac{120}{2}=60$