Finding Distance: Tangents, Circles, And Calculations

Oct 15, 2025 by Blender 54 views

Finding the Distance Between Points A and B

Hey guys! Today, we're diving into a cool geometry problem. We're going to figure out the distance between two points, A and B. The twist? We're dealing with tangents to a circle! Don't worry, it's not as scary as it sounds. This is the kind of problem that's perfect for brushing up on your geometry skills and understanding how different concepts fit together. The question says, "Qual é a distância entre os pontos A e B, sabendo que os segmentos PA e PB são tangentes à circunferência de centro O, onde PO = 25 cm e PA = 24 cm?" or in English, "What is the distance between points A and B, knowing that segments PA and PB are tangent to the circle with center O, where PO = 25 cm and PA = 24 cm?" Let's break it down step by step. This is a classic geometry problem that uses the properties of circles and tangents, and the Pythagorean theorem to arrive at the solution. So, what do we know? We know that PA and PB are tangent to a circle, meaning they touch the circle at only one point. Also, we are given the values of PO and PA and asked to find the length of AB. Given the following options:

A) 30 cm B) 48 cm C) 50 cm D) 52 cm

Let's explore the concepts, work through the solution, and then find the best answer among the given alternatives. Let's get started! We'll unravel the geometry, solve the equation, and get the answer! So grab a pen and paper, and let's get started. Believe me, by the end of this article, you'll not only know the answer, but you'll also have a much better understanding of the underlying geometry concepts. This is going to be fun, right?

Understanding the Geometry: Key Concepts

Alright, before we jump into the calculations, let's make sure we're all on the same page about the geometry. Imagine a circle sitting right in front of you. Now, picture a point outside that circle, let's call it P. From this point P, we can draw two lines that just kiss the circle – these are called tangents. These lines PA and PB are tangents to the circle. The crucial thing to remember about tangents is that they always meet the radius of the circle at a right angle (90 degrees). So, if you draw a line from the center of the circle (O) to the point where the tangent touches the circle (A or B), you create a right angle. This, my friends, is where the Pythagorean theorem comes into play. Now, the problem also gives us some critical information: PO = 25 cm and PA = 24 cm. PO is the distance from the external point P to the center of the circle. PA is the length of one of the tangents. The first thing to note is that PA and PB will have the same length because they are tangents drawn from the same external point to the same circle. This means we've got two right-angled triangles in our scenario: ΔPAO and ΔPBO. In both triangles, we know the hypotenuse (PO) and one side (PA). This information will be invaluable as we proceed to solve the problem. To make sure we are good, let's recap the key concepts:

Tangents: Lines that touch a circle at a single point. They are perpendicular to the radius at the point of contact.
Right Angles: Formed where the tangents meet the radii.
Pythagorean Theorem: Helps to calculate sides in right-angled triangles (a² + b² = c²).
Congruent Triangles: Triangles are congruent if all corresponding sides are equal. In this case, ΔPAO and ΔPBO. That is, PA = PB

Knowing these concepts is important. Next, we'll use them to find the answer. Now that we've laid the foundation, let's begin solving the problem.

Step-by-Step Solution: Finding the Distance AB

Now that we have a good grasp of the geometry, it's time to get our hands dirty with some calculations. Remember the Pythagorean theorem (a² + b² = c²)? We will use it to find the length of OA (or OB), which is also the radius of the circle. Then, we can apply the Pythagorean theorem to the right triangle ΔPAO. In this triangle, PO is the hypotenuse (25 cm), and PA is one of the sides (24 cm). Let's denote the radius of the circle, which is OA, as r. Applying the Pythagorean theorem to ΔPAO, we have:

PA² + OA² = PO² 24² + r² = 25² 576 + r² = 625 r² = 625 - 576 r² = 49 r = √49 r = 7 cm

So, the radius of the circle (OA) is 7 cm. Great! Now, let's consider the distance between A and B. You see, the line segment AB is bisected by the line segment PO. This means that PO cuts AB into two equal parts. Consider ΔPAO and ΔPBO, both are right-angled triangles. Thus, if we draw a line from O that is perpendicular to AB, it will intersect AB at its midpoint, let's call it M. The figure created is also symmetrical about PO, making triangles ΔAMO and ΔBMO also right-angled triangles. Consider ΔAMO. We know OA (the radius = 7 cm) and PA (24 cm). Using the Pythagorean theorem, we get:

AM² + OM² = OA² AM² = OA² - OM²

However, to find the length of AB, we actually need to calculate the length of AM and double it because M is the midpoint of AB. Consider ΔPAO, we know PA = 24 cm, OA = 7cm and PO = 25 cm. The area of the triangle can be calculated using the formula, Area = 1/2 * base * height. Area of ΔPAO = 1/2 * PA * OA

To calculate the area using another way, first calculate the area using Heron's formula. First, we will calculate the semi-perimeter, s, which is:

s = (PA + OA + PO)/2 s = (24 + 7 + 25)/2 s = 56/2 = 28 cm

Then, the area of the triangle is √[s(s - PA)(s - OA)(s - PO)]

Area of ΔPAO = √[28(28 - 24)(28 - 7)(28 - 25)] Area of ΔPAO = √[28 * 4 * 21 * 3] Area of ΔPAO = √[7056] = 84 cm²

From this value of the area, we can calculate AM, using another way. In the triangle ΔPAO, the base is PO, and the height is AM, which is what we want to calculate. Thus:

Area of ΔPAO = 1/2 * PO * AM AM = (2 * Area of ΔPAO) / PO AM = (2 * 84) / 25 AM = 168 / 25 = 33.6 cm

So, since AM = 33.6 cm, and M is the midpoint of AB:

AB = 2 * AM AB = 2 * 33.6 AB = 48 cm.

Therefore, the distance between A and B is 48 cm.

Choosing the Correct Answer

Based on our calculations, the correct answer is:

B) 48 cm

Final Thoughts

And there you have it, guys! We have calculated the distance between points A and B! We broke down a geometry problem step-by-step, using the Pythagorean theorem and some clever geometric insights. This is a classic example of how different mathematical concepts work together to solve a problem. I hope this detailed explanation helped clarify everything. Keep practicing these problems, and you'll become a geometry whiz in no time! If you have any questions, feel free to ask! Happy calculating!