Geometry Problems: Circle Angles & Pyramid Volume

October 15, 2025 by TextBrain Team 50 views

Circle Geometry: Finding Angle ACB

Alright, geometry lovers, let's dive into our first problem! We've got a circle with its center labeled as O. Inside this circle, we have two diameters, AC and BD. Remember, a diameter is a line segment that passes through the center of the circle and connects two points on the circle. We are given that the central angle AOD is 130 degrees. Our mission, should we choose to accept it, is to find the measure of angle ACB. So guys ready?

Understanding the Problem

Before we jump into calculations, let's break down what we know and what we need to find. AC and BD being diameters means that point O is the midpoint of both segments. This also implies that angles AOD and BOC are vertical angles. Vertical angles are always equal, so angle BOC is also 130 degrees. Now, here's where it gets interesting. Angle ACB is an inscribed angle, meaning its vertex (C) lies on the circle, and its sides (CA and CB) are chords of the circle.

Applying the Inscribed Angle Theorem

The inscribed angle theorem is our best friend here. It states that the measure of an inscribed angle is half the measure of its intercepted arc. In this case, angle ACB intercepts arc AB. To find the measure of arc AB, we need to relate it to the central angle that intercepts the same arc. Notice that angle AOB intercepts arc AB. Angles AOD and AOB are supplementary, meaning they add up to 180 degrees because they form a straight line (diameter BD). Therefore, angle AOB = 180° - 130° = 50°.

Calculating Angle ACB

Since the central angle AOB is 50 degrees, the measure of arc AB is also 50 degrees (the measure of a central angle is equal to the measure of its intercepted arc). Now, applying the inscribed angle theorem, the measure of angle ACB is half the measure of arc AB. Therefore, angle ACB = 50° / 2 = 25°. So, there we have it!

Final Answer

The measure of angle ACB is 25 degrees. This problem highlights the importance of understanding the relationships between central angles, inscribed angles, and intercepted arcs in circle geometry. Keep these concepts in your back pocket; they'll come in handy!

Pyramid Volume: A Regular Square Pyramid Challenge

Now, let's switch gears and tackle a 3D geometry problem! We're dealing with a regular square pyramid. What does "regular" mean in this context? It means that the base of the pyramid is a square, and the apex (the top point) of the pyramid is directly above the center of the square base. We are given that the side length of the base is 4, and the length of a lateral edge (the edge connecting a vertex of the base to the apex) is $\sqrt{17}$ . Our goal is to find the volume of this pyramid. This should be fun for you guys.

Understanding the Regular Square Pyramid

Visualizing the pyramid is key. Imagine a square lying flat, and then imagine a point hovering directly above the center of that square. Connect that point to each of the four corners of the square, and you've got a regular square pyramid. The volume of any pyramid is given by the formula: V = (1/3) * Base Area * Height. We know the side length of the square base, so calculating the base area is easy. The challenge lies in finding the height of the pyramid.

Finding the Height of the Pyramid

To find the height, we need to use the information about the lateral edge. Imagine a right triangle inside the pyramid. One leg of this triangle is the height of the pyramid (let's call it h). Another leg is half the length of the diagonal of the square base. The hypotenuse of this triangle is the lateral edge, which we know is $\sqrt{17}$ . First, let's find the length of the diagonal of the square base. If the side length of the square is 4, then the diagonal is 4 $\sqrt{2}$ (using the Pythagorean theorem or the property of a 45-45-90 triangle). Therefore, half the length of the diagonal is 2 $\sqrt{2}$ .

Now we can apply the Pythagorean theorem to our right triangle: h² + (2 $\sqrt{2}$ )² = ( $\sqrt{17}$ )². This simplifies to h² + 8 = 17. Subtracting 8 from both sides, we get h² = 9. Taking the square root of both sides, we find that h = 3. So, that's cool! We found out what h is, and it's 3.

Calculating the Volume

Now that we know the height, we can calculate the volume of the pyramid. The base area is simply the area of the square, which is side * side = 4 * 4 = 16. Plugging this into the volume formula, we get: V = (1/3) * 16 * 3 = 16. Boom! We just calculated the volume.

Final Answer

The volume of the regular square pyramid is 16 cubic units. This problem demonstrates how to combine geometric properties with the Pythagorean theorem to solve for unknown dimensions and ultimately calculate the volume of a 3D shape. Always visualize the problem and break it down into simpler components. That's the coolest thing you can do!

These problems showcase how understanding key geometric principles and theorems can help solve complex problems. Keep practicing, and you'll become a geometry master in no time!