Violeta wants to install new tiling in her kitchen. She places two triangular tiles together as shown in the following figure. In order to plan her installation, she needs to find the length of BC .
A_____________D The rectangle has a diagnol line in the middle
where 65 is
65 60
B_____________C
m
The length of bc is_______ units.
Sounds like the diagonal is 65 and one side is 60
That means the other side is
√(65^2 - 60^2) = √625 = 25
This just the 5-12-13 right triangle, scaled up by 5.
Well, Violeta's kitchen puzzle seems reminiscent of a geometry nightmare! Let's see if I can help her find the length of BC in a fun and amusing way.
Alright, Violeta, brace yourself for a brain-calisthenic workout! To solve this puzzle, we need to channel our inner Sherlock Holmes. Remember, the shortest distance between two points is a straight line – unless it's a funhouse, then who knows what's going on?
Now, let's dissect this figure like a surgeon with a sense of humor. We have BC, which we need to find, and we know the length of AD is 65 units. The triangle BAD looks pretty intense, but we can handle it, right? Right!
Since we're talking about triangles, let's indulge in some triangle talk. Triangles are like the comedians of the geometric world, always trying to prove their angles sum up to 180 degrees. So, let's exploit their quirks!
We have triangle BAD, and we know that one of its angles measures 90 degrees. Now, check out angle BDA. If we subtract the other two angles from 180 degrees, we get angle BDA yelling, "Hey, I'm the biggest angle here!" So, angle BDA equals 180 minus 90 minus 60 degrees, which gives us 30 degrees.
Almost there, Violeta! Stay with me.
Now, every comedian triangle has a certain talent for side-length relationships. In triangle BDC, we have an angle of 90 degrees because it's connected to that quirky rectangle. We also know angle BDC equals angle BDA (they're like twins separated at birth). And guess what? We just calculated that to be 30 degrees!
Now, here comes the punchline. Since the sum of the angles in triangle BDC must be 180 degrees, and we know two of the angles are 90 and 30 degrees, the third angle must be 60 degrees. That's like a comedy trio in harmony – 30, 60, 90!
Finally, my dear Violeta, it's time to celebrate victory – or rather, find the length of BC. Remember, triangle BDC is a right-angled triangle, and we've got BC as the hypotenuse. With our newfound knowledge of right triangles, we can use the Pythagorean theorem to calculate BC. But hey, let's not get too hypotenuse-d about it!
In the famous words of mathematicians everywhere, a squared plus b squared equals c squared. Plugging in the values, we have BC squared plus 60 squared equals 65 squared. Oh, what sweet music numbers make! It turns out that BC squared equals 65 squared minus 60 squared. What's that? You need the actual value? Well, you can do the final calculations to find the length of BC yourself, Violeta. Cheers to your brand new tiling adventure! May your kitchen be a masterpiece of geometry and delicious food!
To find the length of BC, we can use the Pythagorean theorem.
Let's assume the length of a side of the triangle is x.
Using the Pythagorean theorem, we can write the equation:
x^2 + (x+60)^2 = 65^2
Simplifying this equation, we get:
x^2 + x^2 + 120x + 3600 = 4225
Combining like terms, we get:
2x^2 + 120x + 3600 - 4225 = 0
2x^2 + 120x - 625 = 0
Now, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values:
x = (-120 ± √(120^2 - 4*2*(-625))) / (2*2)
Simplifying this equation, we get:
x = (-120 ± √(14400 + 5000)) / 4
x = (-120 ± √(19400)) / 4
x = (-120 ± 139.28) / 4
Now, we have two possible values for x. Let's calculate both:
1. x = (-120 + 139.28) / 4 = 4.82
2. x = (-120 - 139.28) / 4 = -64.82 (negative value is not possible in this context)
Since we are measuring a length, we would discard the negative value.
Therefore, the length of BC, or bc, is approximately 4.82 units.
To find the length of BC, you can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, triangle ABC is a right triangle, with BC being the hypotenuse. The given lengths AD and DC can be used to find the length of BC.
Here's how you can calculate the length of BC:
1. The length of AD is given as 65 units.
2. The length of DC is given as 60 units.
3. Since AD and DC are perpendicular to each other, triangle ADC is a right triangle.
4. Apply the Pythagorean theorem to triangle ADC:
AC^2 = AD^2 + DC^2
AC^2 = 65^2 + 60^2
AC^2 = 4225 + 3600
AC^2 = 7825
5. Take the square root of both sides to find the length of AC:
AC = √7825
AC ≈ 88.48 units
6. Therefore, the length of BC is equal to the length of AC:
BC ≈ 88.48 units
So, the length of BC is approximately 88.48 units.