Select The Correct Answer From Each Drop-down Menu.Given: $\triangle ABC$ With Altitude $h$. Two Right Triangles Are Formed: One With Side Lengths $c + R$, $h$, And $b$, And One With Side Lengths

**Understanding the Properties of Right Triangles in $\triangle ABC$**

In the world of geometry, right triangles play a crucial role in understanding various mathematical concepts. When dealing with right triangles, it's essential to grasp the properties and relationships between their sides. In this article, we will delve into the properties of right triangles formed within $\triangle ABC$ with altitude $h$. We will explore the relationships between the sides of these triangles and provide a comprehensive understanding of the given problem.

Given: $\triangle ABC$ with altitude $h$. Two right triangles are formed: one with side lengths $c + r$, $h$, and $b$, and one with side lengths $c$, $r$, and $h$. The task is to select the correct answer from each drop-down menu.

To approach this problem, we need to understand the properties of right triangles and the relationships between their sides. Let's start by analyzing the two right triangles formed within $\triangle ABC$.

Triangle 1: $(c + r)$, $h$, and $b$

The first right triangle has side lengths $c + r$, $h$, and $b$. We can see that the side length $c + r$ is the hypotenuse of this triangle, while $h$ is one of the legs. The side length $b$ is the other leg of this triangle.

Triangle 2: $c$, $r$, and $h$

The second right triangle has side lengths $c$, $r$, and $h$. In this triangle, $c$ and $r$ are the legs, while $h$ is the hypotenuse.

Now that we have a clear understanding of the two right triangles, let's explore the key relationships between their sides.

Relationship between $c$ and $r$

We know that $c$ and $r$ are the legs of the second right triangle. Since this is a right triangle, we can apply the Pythagorean theorem, which states that the sum of the squares of the legs is equal to the square of the hypotenuse.

$$c^2 + r^2 = h^2$$

Relationship between $c + r$ and $h$

In the first right triangle, $c + r$ is the hypotenuse, while $h$ is one of the legs. We can apply the Pythagorean theorem again to find the relationship between $c + r$ and $h$.

$$(c + r)^2 = h^2 + b^2$$

Relationship between $c$, $r$, and $b$

We can also find the relationship between $c$, $r$, and $b$ by applying the Pythagorean theorem to the first right triangle.

$$c^2 + b^2 = (c + r)^2$$

Relationship between $c$, $r$, and $h$

Finally, we can find the relationship between $c$, $r$, and $h$ by applying the Pythagorean theorem to the second right triangle.

$$c^2 + r^2 = h^2$$

In conclusion, we have explored the properties of right triangles formed within $\triangle ABC$ with altitude $h$. We have analyzed the relationships between the sides of these triangles and provided a comprehensive understanding of the given problem. By applying the Pythagorean theorem, we have found the key relationships between the sides of the triangles.

Q: What is the relationship between $c$ and $r$?

A: The relationship between $c$ and $r$ is given by the Pythagorean theorem: $c^2 + r^2 = h^2$.

Q: What is the relationship between $c + r$ and $h$?

A: The relationship between $c + r$ and $h$ is given by the Pythagorean theorem: $(c + r)^2 = h^2 + b^2$.

Q: What is the relationship between $c$, $r$, and $b$?

A: The relationship between $c$, $r$, and $b$ is given by the Pythagorean theorem: $c^2 + b^2 = (c + r)^2$.

Q: What is the relationship between $c$, $r$, and $h$?

A: The relationship between $c$, $r$, and $h$ is given by the Pythagorean theorem: $c^2 + r^2 = h^2$.

Based on the analysis and relationships between the sides of the triangles, we can conclude that the correct answer is:

• For the first drop-down menu: $(c + r)^2 = h^2 + b^2$
• For the second drop-down menu: $c^2 + r^2 = h^2$

Note: The final answer may vary depending on the specific values of $c$, $r$, and $h$.