## Concept explanation **Pythagoras’ theorem** connects the three side lengths of a triangle only when the triangle contains a **right angle** of `90°`. In that special case, the two shorter sides and the longest side satisfy `a² + b² = c²`. If the triangle does **not** have a right angle, that relationship no longer has to be true, so the theorem does not apply. ## What you see You are looking at one triangle that can switch between two shapes. The corner at point `B` is highlighted with an angle marker, and the status card tells you whether the theorem applies in the current shape. When the triangle becomes non-right, the angle label changes and the status updates to show that the theorem depends on having a `90°` corner. ## Try it yourself - **Click the toggle button** to switch from a right triangle to a non-right triangle and watch the status change from `Applies: yes` to `Applies: no`. - **Switch back and forth a few times** and notice that the theorem only returns when the marked corner becomes `90°` again. - **Turn the angle marker on and off** with the checkbox to focus either on the shape change or on the angle cue. - **Compare the two states** and notice that the key difference is not just the triangle’s appearance, but whether one angle is exactly `90°`. ## Concept explanation In a **right triangle**, the two sides that meet to make the **right angle** are called the **legs**. The side directly across from that `90°` corner is always the **hypotenuse**. The hypotenuse is also the **longest side**, so you can identify it by finding the side opposite the right angle rather than just guessing from its slant or position. ## What you see You’re looking at a color-coded right triangle with one corner marked `90°`. Each side has its own color, and when you hover over a side, the display names that side and explains its role. The information card keeps pointing out that the side opposite the right-angle corner is the hypotenuse, while the other two sides are the legs. ## Try it yourself - **Hover over each colored side** and compare the name that appears. - **Find the `90°` corner first** and then look across from it to predict the hypotenuse before hovering. - **Change the triangle size** with the slider and notice that the hypotenuse is still the side opposite the right angle. - **Switch between `Standard`, `Tall`, and `Wide`** to see that the triangle can change shape, but the hypotenuse rule stays the same. - **Press `Reset`** and test yourself again by identifying the two legs and the hypotenuse. ## Concept explanation For any **right triangle**, you can build a **square** outwardly on each of its three sides. The two shorter sides are the **legs**, and the longest side is the **hypotenuse**. Because a square’s **area** is its side length multiplied by itself, each square’s area changes as that triangle side changes. In a right triangle, the square on the hypotenuse grows from the changing leg lengths, which is the geometric idea behind the Pythagorean relationship. ## What you see You’re looking at a right triangle with one attached square on each side. The blue square is built on side `a`, the green square on side `b`, and the gold square on side `c`. The triangle keeps its 90° corner while the sliders change the leg lengths, and the numbers inside the squares update so you can compare side length and area at the same time. ## Try it yourself - **Move the base-leg slider** and watch the blue square stretch or shrink as side `a` changes. - **Move the vertical-leg slider** to change side `b` while keeping the triangle a right triangle. - **Compare the area readouts** in the control panel with the labels inside each square. - **Notice how larger side lengths create much larger square areas**, because area depends on length squared. - **Press `Reset`** to return to an equal-leg right triangle and compare the three attached squares again. ## Concept explanation In a right triangle, the two shorter sides are called the **legs** and the longest side is the **hypotenuse**. If you build a square on each side, the area of the square on one leg plus the area of the square on the other leg is exactly equal to the area of the square on the hypotenuse. This is the **Pythagorean theorem**: `a² + b² = c²`. The animation shows that this is not just a formula to memorize — the smaller areas can be rearranged so they completely cover the larger square. ## What you see You are looking at a right triangle with three attached squares labeled `a²`, `b²`, and `c²`. The blue and green tiles represent area coming from the two smaller squares, while the gold square sits on the hypotenuse. A faint divider inside the `c²` square marks how the hypotenuse square is partitioned so the blue area and green area together fill it with no gaps and no overlaps. ## Try it yourself - **Click `Animate rearrangement`** and watch the blue and green area pieces move into the `c²` square. - **Notice** that the pieces from `a²` fill one part of the hypotenuse square while the pieces from `b²` fill the rest. - **Click `Reset`** and replay the motion to check that the final filled area is always exact. - **Drag the `Grid density` slider** to change how many pieces the squares are cut into, then animate again. - **Toggle `Show piece grid`** to simplify the view or to inspect the area partition more closely. - **Compare** the final filled `c²` square with the starting `a²` and `b²` squares, and confirm that the combined area stays the same during the rearrangement. ## Concept explanation In a right triangle, the **Pythagorean theorem** says that the two shorter sides, called the **legs**, satisfy `a² + b² = c²`, where `c` is the **hypotenuse**, the side opposite the right angle. You can use this relationship to find a missing side: if the unknown side is the hypotenuse, add the squares of the two legs and take the square root; if one leg is unknown, subtract the known leg’s square from the hypotenuse’s square and then take the square root. ## What you see You’re looking at a right triangle with one side fixed at `a = 9` and another side controlled by the slider. The live equation card updates as the triangle changes, showing the substitution step and the final square-root calculation for the unknown side. You can also switch between solving for the hypotenuse and solving for the other leg, so you can see both common ways the theorem is used. ## Try it yourself - **Move the known-side slider** and watch how the triangle stretches while the equation updates at the same time. - **Keep the mode on “Find hypotenuse”** and notice that increasing either leg makes the hypotenuse grow. - **Switch to “Find other leg”** to see how the theorem is rearranged to `b = √(c² - a²)`. - **Compare small and large values** to see how squaring changes the numbers before the square root brings the result back to a side length. - **Drag the highlighted vertical side** when finding the hypotenuse to change the same value directly on the diagram. ## Concept explanation For any **right triangle**, the two shorter sides are called the **legs** and the longest side is the **hypotenuse**. **Pythagoras’ theorem** says that if the leg lengths are `a` and `b`, and the hypotenuse is `c`, then `a² + b² = c²`. This is not just true for one special triangle like `3-4-5`—it stays true for every right triangle, no matter how stretched or squashed it becomes, as long as the angle stays `90°`. ## What you see You can move the teal corner to create many different right triangles while the right angle stays fixed at the white corner. The colored squares are built on the three sides, and their numerical areas update live. The measurement panel shows `a²`, `b²`, `c²`, and the tiny difference between `a² + b²` and `c²`, so you can watch the equality hold continuously as the triangle changes. ## Try it yourself - **Drag the teal point** to make the triangle taller, wider, or both, and watch all three side lengths update. - **Compare the numbers** for `a²`, `b²`, and `c²` in the measurement panel as you move the point. - **Notice that `a² + b²` always matches `c²`** even though the triangle’s shape changes. - **Choose a preset triangle** like `3-4-5` or `5-12-13` to see familiar whole-number examples. - **Adjust the base leg target slider** and observe how changing one leg affects the hypotenuse and all three square areas. - **Toggle “Show squares”** to focus either on the geometric areas or just on the triangle and live numerical relationship.