Calculate the Perimeter of Triangle ACD in a Square with Diagonal

Question

Observe the square below:XXXAAABBBDDDCCC

Determine the perimeter of the triangle ACD?

Video Solution

Step-by-Step Solution

In order to answer the question, we first need to recall the properties of a square.

In a square, all the sides are equal. Therefore, the lengths of all sides, AB AB , BD BD , DC DC and AC AC are equal.
Since we denoted AC=X AC=X , we can state that:

AB=BD=DC=AC=X AB=BD=DC=AC=X

Now let's remember another property of a square, which is that in a square all angles are equal to 90 90 degrees.
This means that triangle ACD ACD is a right triangle, because angle C (which is part of the square) equals 90 90 degrees.

In a right triangle, we can use another tool we have - the Pythagorean theorem.
The Pythagorean theorem allows us, in a right triangle, to determine the length of the third side using the other two sides as shown below:

A2+B2=C2 A²+B²=C²

C is the hypotenuse.

Therefore we can insert the given data:

AC2+CD2=AD2 AC²+CD²=AD²

Since we know that AC=CD=X AC=CD=X , the equation is as follows:

X2+X2=AD2 X²+X²=AD²

2X2=AD2 2X²=AD²

2X2=AD \sqrt{2X²}=AD

2X=AD \sqrt2 \cdot X=AD

Thus we have found the third side of the triangle.

However we're not done yet!

Remember, we were asked to find the perimeter of triangle ACD,
Remember, the perimeter of a triangle is the sum of its sides.

AC+CD+AD=perimeter AC+CD+AD= perimeter

X+X+2X=perimeter X+X+\sqrt2 \cdot X=perimeter

2X+2X=perimeter 2X+\sqrt2 X = perimeter

That's the solution!

Answer

2x+2x 2x+\sqrt{2}x


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