In order to build a deck at a right angle to their house, Lucinda and Felipe place a stake in th

step 1 So this is a word problem involving a right angle. The word problem itself, after you read throu

In order to build a deck at a right angle to their house,...

Key Concepts

Consecutive Integers

Integers that follow one another in order, with a difference of one between each pair. In problems with numerical constraints, they require finding numbers that not only meet sequence requirements but also fulfill additional conditions, such as satisfying the Pythagorean theorem.

Right Triangle

A triangle with one 90-degree angle, which serves as a basis for many geometric constructions and proofs. Understanding the properties of right triangles is essential for solving problems that involve determining unknown side lengths using methods like the Pythagorean theorem.

Pythagorean Theorem

A fundamental principle in geometry that states in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is crucial in determining relationships between the sides of a right triangle and is used to verify if given side lengths can form a right triangle.

Pythagorean Triple

A set of three positive integers that satisfy the Pythagorean theorem, meaning they can represent the side lengths of a right triangle. Recognizing such triples can simplify calculations and provide quick insights into problems involving right triangles.

Transcript

00:01 So this is a word problem involving a right angle.

00:05 The word problem itself, after you read through it, outlines the looks of the right triangle that is to be created.

00:15 So when reading the word problem, you are given a leg right here that will be called x.

00:24 Consecutive integers will create this triangle.

00:28 And consecutive integers, we know 3, 4, 5, 6, 7, those are consecutive.

00:32 Each number is one greater than the number before it.

00:38 So there's the first consecutive integer after x.

00:44 And then the next consecutive integer after x will be an additional one after.

00:50 And there then becomes my hypotenuse.

00:52 How do i know that's my hypotenuse? the green one? i know that because it is the larger of the three numbers.

01:00 Good practice.

01:02 Wrap any by.

01:03 I know meals in parentheses.

01:05 Whenever you are going to have to square those, it becomes much more complex than it looks.

01:13 My right angle is right here.

01:19 So the question is, what are the dimensions of the three legs? so we will have three answers when we're finished.

01:27 Let us go to pythagorean serum, which states that the square sum, which is an addition sign, of the square of the two legs, which we will call a and b, will always be equal to the square of the hypotenuse, which is always c.

01:44 Now, a and b don't matter which order you put in the two legs.

01:49 X is one leg, x plus one is the second leg.

01:54 C does matter.

01:55 It must be the x plus two.

01:58 So when i read left to right, the first leg i see is the x.

02:01 So i'm going to substitute the x in for the a, and i get x squared.

02:07 I'm going to substitute in parentheses, x plus 1 in for b, and i will get the binomial squared.

02:17 This is not as simple as squaring the x and squaring the 1 and calling it x squared plus 1.

02:24 It is not that simple.

02:26 So please remember that.

02:28 We'll get to the foiling of that in just a moment.

02:32 C is x plus 2.

02:33 Again, in parentheses, this is not as simple as x squared plus 4.

02:41 Okay, now my job is to solve for x, but i first must remove all the parentheses.

02:46 And to remove the parentheses, i have to square the binomial.

02:51 So the first term remains x squared.

02:56 I'm going to write out in red everything that comes out of the x plus one.

03:00 We get the x plus, i'm sorry, the x times x, which is x squared, plus x times one, which is one x, times one plus x, which is another one x, plus 1 times 1.

03:16 That is x plus 1 squared right there.

03:18 Four terms.

03:20 Now i'm going to do the same to my x plus 2.

03:23 X times x is x squared.

03:28 X times 2 is 2x.

03:31 2 times x is 2x.

03:35 2 times x is 2x and 2 times 2 is 4.

03:41 There we go.

03:43 So that is my binomial squared being very careful with the squaring of.

03:49 Them.

03:50 Now what i'm going to do is combine like terms on either side.

03:55 What i do see is x squared plus x squared can be combined to become 2x squared.