Parts of a Triangle: Finding the ratio between dimensions in a triangle

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

• Step 1: Recognize that triangle ABC is a right triangle with BD as the median to the hypotenuse AC.
• Step 2: Apply the geometric property that the median to the hypotenuse of a right triangle is half the length of the hypotenuse.

Now, let's work through these steps:
Step 1: The problem states that BD is the median to the hypotenuse AC in the right triangle ABC.
Step 2: According to the geometric property of a median in a right triangle, the length of the median BD is half the hypotenuse $AC$. Therefore, $BD = \frac{1}{2}AC$.

Thus, the ratio between $BD$ and $AC$ can be expressed as:
$\frac{BD}{AC} = \frac{\frac{1}{2}AC}{AC} = \frac{1}{2}$
This simplifies to the ratio $1:2$.

Therefore, the solution to the problem is $1:2$.

In triangle $ABC$, $AD$ is a median from $A$ to side $BC$. By definition, a median divides the opposite side into two equal segments: $BD = DC$. Therefore, the total length of side $BC$ is the sum of $BD$ and $DC$, which simplifies to $BC = BD + DC = 2 \times DC$.

We are tasked with finding the ratio $BC : DC$. Given that $BC = 2 \times DC$, we conclude the ratio is $2:1$.

Therefore, the solution to the problem is $2:1$.

To solve this problem, we'll focus on the role of the median $AD$.

• Since $AD$ is a median, it divides side $BC$ into two equal parts, meaning $D$ is the midpoint.
• The property of a median in a triangle is that it divides the triangle into two smaller triangles of equal area.
• Thus, $\triangle ABD$ and $\triangle ADC$ will have the same area.

Therefore, the ratio of the area of $\triangle ABD$ to the area of $\triangle ADC$ is $1:1$.