Trimming a gate. A total of 34 feet of 1 ×4 lumber is used around the perimeter of the gate show

step 1 All right, on this problem, you're told that you have a gate. The diagonal of it is 13. And so w

Trimming a gate. A total of 34 feet of 1 ×4 lumber is used...

Trimming a gate. A total of 34 feet of $1 \times 4$ lumber is used around the perimeter of the gate shown in the figure. If the diagonal brace is 13 feet long, then what are the length and width of the gate? (GRAPH CANNOT COPY).

Key Concepts

Systems of Equations

Solving problems that involve both a perimeter equation and a diagonal relation often requires setting up a system of simultaneous equations. This method involves using more than one equation to find the values of the unknown variables while ensuring that both the perimeter and the relationship imposed by the Pythagorean theorem are satisfied.

Quadratic Equations

The process of solving a system that includes a squared term (originating from the Pythagorean theorem) typically leads to a quadratic equation. Understanding how to factor or use the quadratic formula is essential for finding the correct dimensions that satisfy both the perimeter and the diagonal conditions.

Perimeter of a Rectangle

This concept involves calculating the total distance around a rectangle, which is found by summing twice its length and twice its width. It is a fundamental aspect of geometry used to relate the linear dimensions of a rectangle to a given total boundary length.

Pythagorean Theorem

The Pythagorean Theorem is a key principle in geometry that relates the sides of a right triangle. In the context of a rectangle, the diagonal forms a right triangle with the length and width as its legs, enabling the calculation of one dimension if the other and the diagonal length are known.

Transcript

00:01 All right, on this problem, you're told that you have a gate.

00:03 The diagonal of it is 13.

00:05 And so with pythagorean theorem here, that tells us x squared plus y squared is equal to 13 squared.

00:11 13 squared is 169.

00:15 And then you're also told that around the gate, you're going around the gate here, takes 34 feet of lumber.

00:22 And so that means that would be 2x plus 2y would be 34, because you've got two sides of those x's, two sides of those ys.

00:32 Now if you divide it, this all by 2 here, that would tell us x plus y is equal to 17.

00:39 And if you take the x over, that tells us y is 17 minus x.

00:45 And the whole reason i do that is so that i can plug this thing in up there for y.

00:52 And so it tells us we have x squared plus 17 minus x squared equals 16.

01:02 For 17 minus x squared, you've got to take 17 minus x, multiply it by itself.

01:07 Okay, foil it out.

01:11 When you do that, that's going to give you 289 minus 34x plus x squared is equal 169.

01:20 I'll let you verify that, all right, but you multiply it by itself and then foil it out.

01:24 And so now we have this quadratic equation to solve here.

01:30 Now to solve this quadratic, let's combine our like terms.

01:32 So x squared plus x squared is 2x squared.

01:37 Minus 34x.

01:42 And then i need to get it equal to zero.

01:43 So let's subtract 169 over...

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