1. **Problem:** Label the sides of a right triangle with vertices P (top), Q (bottom-left, right angle), and R (bottom-right). State the Pythagorean theorem as it applies to these sides. 2. **Problem:** A ladder $l$ is placed against a house of height $h$. The foot of the ladder is a distance $d$ from the base of the house. Express the relationship between $l$, $h$, and $d$. 3. **Problem:** Rearrange the Pythagorean theorem $x^2 + y^2 = z^2$ to solve for $x$ and then for $y$. --- ### Step 1: Labeling the triangle and stating the theorem - In a right triangle, the side opposite the right angle is the hypotenuse. - Here, vertex $Q$ is the right angle. - Therefore, side $PR$ is the hypotenuse. - Sides $PQ$ and $QR$ are the legs. The Pythagorean theorem states: $$\text{(leg)}^2 + \text{(leg)}^2 = \text{(hypotenuse)}^2$$ For triangle $PQR$: $$PQ^2 + QR^2 = PR^2$$ --- ### Step 2: Relationship between ladder $l$, height $h$, and distance $d$ - The ladder $l$ forms the hypotenuse of a right triangle. - The height of the house $h$ is one leg. - The distance from the base $d$ is the other leg. By the Pythagorean theorem: $$d^2 + h^2 = l^2$$ --- ### Step 3: Rearranging the Pythagorean theorem $x^2 + y^2 = z^2$ - To solve for $x$: $$x^2 = z^2 - y^2$$ $$x = \sqrt{z^2 - y^2}$$ - To solve for $y$: $$y^2 = z^2 - x^2$$ $$y = \sqrt{z^2 - x^2}$$ --- **Final answers:** 1. $PQ^2 + QR^2 = PR^2$ 2. $d^2 + h^2 = l^2$ 3. $x = \sqrt{z^2 - y^2}$ and $y = \sqrt{z^2 - x^2}$