Trapez Prostokątny: Oblicz Wysokość I Przekątną
Hey math whizzes! Today, we're diving deep into the world of trapezoids, specifically the right trapezoid. You know, the one with those nice, handy right angles? We've got a super common problem to tackle, where we need to figure out the height and the longer diagonal. It's all about using what we know and applying some classic geometric principles. So, grab your notebooks, and let's break down this problem step-by-step.
Understanding Our Right Trapezoid
Alright guys, let's get our heads around the problem. We're dealing with a right trapezoid. This means it has at least one pair of parallel sides (those are our bases) and one of the non-parallel sides is perpendicular to both bases. This perpendicular side is what we call the height of the trapezoid. In our specific case, the parallel bases have lengths of 2 and 14. That's a pretty big difference, right? Then, we're given the length of the longer non-parallel side, which is 13. This longer side is NOT the height, mind you, because if it were, it would be perpendicular to the bases. The fact that it's called the longer side suggests there's another non-parallel side, and this longer one is slanted. Our mission, should we choose to accept it, is to find:
- The height (h) of the trapezoid.
- The length of the longer diagonal (d).
This is a classic geometry problem that tests your understanding of Pythagorean theorem and basic properties of geometric shapes. It might seem a bit daunting at first, but trust me, once we break it down, it's totally manageable. Think of it like solving a puzzle. We have pieces of information, and we need to fit them together using the right tools. The key here is to visualize the trapezoid and, more importantly, to draw auxiliary lines that create right-angled triangles. These triangles are our best friends when dealing with lengths and heights in trapezoids, especially right trapezoids.
So, let's start by sketching this bad boy out. Imagine a horizontal line at the bottom (length 14) and a shorter one on top (length 2), parallel to it. Then, draw a vertical line on one side connecting the ends of the bases – that's our height. Now, connect the other ends of the bases with a slanted line – that's our longer non-parallel side (length 13). The challenge is that we don't immediately know the height. But, here's a common trick: if we draw a line parallel to the height from the end of the shorter base to the longer base, what do we get? Yep, another right-angled triangle! This little maneuver is going to be crucial for finding our missing height. We'll be using the Pythagorean theorem, a² + b² = c², where 'c' is always the hypotenuse (the longest side, opposite the right angle). Let's get ready to crunch some numbers!
Calculating the Height of the Trapezoid
Alright, let's get down to business and calculate the height of our trapezoid. This is where our trusty geometric tricks come into play. Remember that right trapezoid we visualized? We have bases of lengths 2 and 14, and the longer non-parallel side is 13. The key to finding the height is to create a right-angled triangle within the trapezoid. How do we do that? We draw a line segment from one of the vertices of the shorter base, parallel to the height, down to the longer base.
Let's visualize this. Imagine the trapezoid ABCD, where AB is the shorter base (length 2) and CD is the longer base (length 14). Let AD be the height (perpendicular to both bases). BC is the longer non-parallel side (length 13). Now, draw a line from vertex B perpendicular to the base CD, and let's call the point where it meets CD as E. So, BE is the height we want to find. What have we created? We've formed a rectangle ABED and a right-angled triangle BCE.
Why is this helpful? Because the length of BE is the same as the length of AD, which is our height (h). Also, the length of DE is the same as the length of AB, which is 2. Now, look at the base CD. It's made up of two segments: DE and EC. We know CD = 14 and DE = 2. So, the length of EC is CD - DE = 14 - 2 = 12. Now we have a right-angled triangle, triangle BCE, where:
- BE is the height (h) - this is what we want to find.
- EC is one of the legs, and we found its length to be 12.
- BC is the hypotenuse (the longest side, opposite the right angle at E), and we know its length is 13.
Now, we can finally use the Pythagorean theorem: a² + b² = c². In our triangle BCE:
- a = BE = h
- b = EC = 12
- c = BC = 13
So, the equation becomes: h² + 12² = 13².
Let's solve for h²:
h² + 144 = 169
Subtract 144 from both sides:
h² = 169 - 144
h² = 25
To find h, we take the square root of both sides:
h = √25
h = 5
Boom! We've found it. The height of the trapezoid is 5. See? It wasn't so bad once we drew that extra line and formed our right-angled triangle. This is a super common technique in geometry problems, so remember it. When in doubt, try to create right triangles!
Calculating the Longer Diagonal
Now that we've conquered the height, let's move on to finding the length of the longer diagonal. Remember our trapezoid ABCD, with bases AB=2 and CD=14, height AD=5, and the non-parallel side BC=13. A diagonal connects opposite vertices. We have two diagonals: AC and BD. We need to figure out which one is longer and calculate its length. Usually, in a trapezoid where one non-parallel side is the height, the diagonal that starts from the end of the shorter base not adjacent to the height, and goes to the opposite vertex on the longer base, will be the longer one. In our case, that would be diagonal BD.
Let's focus on finding the length of diagonal BD. To do this, we'll again use the Pythagorean theorem. We need to find a right-angled triangle that includes BD as its hypotenuse. We can create this triangle by extending the height AD downwards and dropping a perpendicular from B to the line extending AD. However, a more straightforward approach involves using the base CD and the height AD along with the segment AB. Consider the right-angled triangle ABD. We know AD = 5 (the height). What about the length of the segment AB? It's the shorter base, which is 2. So, in triangle ABD, AD is one leg (height = 5), and AB is the other leg (length = 2). This is incorrect, as AB is parallel to the base, not perpendicular. My bad!
Let's correct that. We need a right triangle where BD is the hypotenuse. Consider the base CD (length 14). We know that the segment DE has length 2 (from our previous step where we created the rectangle ABED). So, the length of CE is 12. Now, let's think about the diagonal BD. It connects vertex B to vertex D. We can form a right-angled triangle by dropping a perpendicular from B to the longer base CD at point E. We already established that BE = 5 (the height) and EC = 12. However, this triangle (BCE) has BC as the hypotenuse, not BD.
Okay, let's rethink this. We need the diagonal BD. Let's look at the vertices. We have A, B (top base) and C, D (bottom base). Let's assume A is top-left, B is top-right, D is bottom-left, and C is bottom-right. So AB is parallel to DC. AD is perpendicular to DC (height). The length of AB = 2, DC = 14. The non-parallel side is BC = 13. We want to find the longer diagonal. The diagonals are AC and BD. Let's find BD first.
To find BD, we can consider the right-angled triangle formed by the height AD and the segment AB. This doesn't make sense. Let's use coordinates. Let D be at (0,0). Then C is at (14,0). Since AD is the height and it's perpendicular, A is at (0, 5). Since AB is parallel to DC and has length 2, B must be at (2, 5). Now, let's calculate the length of the diagonal BD using the distance formula. The coordinates of B are (2, 5) and D are (0, 0).
Distance BD = √[(x₂ - x₁)² + (y₂ - y₁)²] BD = √[(0 - 2)² + (0 - 5)²] BD = √[(-2)² + (-5)²] BD = √[4 + 25] BD = √29
So, the length of diagonal BD is √29. Now, let's check the other diagonal, AC. The coordinates of A are (0, 5) and C are (14, 0).
Distance AC = √[(x₂ - x₁)² + (y₂ - y₁)²] AC = √[(14 - 0)² + (0 - 5)²] AC = √[14² + (-5)²] AC = √[196 + 25] AC = √221
Comparing √29 and √221, it's clear that √221 is the longer diagonal. So, the length of the longer diagonal is √221.
Self-correction: My previous approach of drawing a line from B to E on CD was for calculating the length of BC when you know the bases and height. For diagonals, coordinates or constructing specific right triangles using the known segments is key. The coordinate method is quite clean here.
Let's re-verify the calculation of the longer diagonal without coordinates, using geometric construction.
Consider the diagonal BD. To find its length, we need a right triangle where BD is the hypotenuse. We can extend AD downwards and draw a perpendicular from B to the line AD extended. This is not practical. Let's use the segment DC and the height AD. We need a point such that we can form a right angle. Let's use the point D and the segment extending from D along the base. We know the length of the base DC is 14. The height AD is 5. Vertex B is located such that AB = 2 and AD = 5. So, imagine a point P directly below B on the line DC. Then DP = AB = 2, and PB = AD = 5. Triangle DPB is a right-angled triangle with legs DP=2 and PB=5. The hypotenuse is DB (diagonal BD).
Using Pythagorean theorem on triangle DPB: BD² = DP² + PB² BD² = 2² + 5² BD² = 4 + 25 BD² = 29 BD = √29
This is the length of one diagonal. Now let's consider the diagonal AC. To find its length, we need a right triangle involving AC. We can drop a perpendicular from A to the base DC. Let's call the point E. Then AE = AD = 5. What is the length of DE? In a right trapezoid where AD is the height, if we draw a perpendicular from B to DC at point E, then DE = AB = 2, and EC = DC - DE = 14 - 2 = 12. Now, consider the right triangle AEC. The legs are AE = 5 and EC = 12. The hypotenuse is AC.
Using Pythagorean theorem on triangle AEC: AC² = AE² + EC² AC² = 5² + 12² AC² = 25 + 144 AC² = 169 AC = √169 AC = 13
Okay, I seem to have made a mistake in my initial assumption or calculation. Let's re-evaluate the problem setup and the meaning of
