Practice Problem 5
What is the perimeter of this figure? The diagonal of the square is the square root of 12. Both of the triangles are equilateral and the length of their altitude is 9.
First, we need to find the lengths of the sides. Let's do the square first. Because each angle is 90 degrees, the diagonal splits the angle into two 45 degree angles. This lets us know that there are two 45-45-90 triangles in the square. By using the ratios of a 45-45-90 triangle, we can figure out that the side length the square root is because:
sqrt12/sqrt2(sqrt2/sqrt2)=sqrt24/2=sqrt12
Next, we need to know the lengths of the triangle. Since the equilateral triangle is divided by the altitude, the altitude splits into two 30-60-90 triangles. From the 30-60-90 Triangle Theorem, we can conclude that the shorter leg is 3sqrt3 because 9/sqrt3(sqrt3/sqrt3)=9sqrt3/3=3sqrt3. Then, we need to find the length of the hypotenuse. To find this, we need to multiply 3sqrt3 by 2 to get 6sqrt3.
The last step is to add all the sides together. For the sake of easier calculations, we can simplify 6sqrt3 to sqrt108. This would give us the equation: 2(sqrt12)+2(sqrt108). Once we solve the equation, we get sqrt240 because sqrt24+sqrt216=sqrt240. When we simplify the square root, we get 4sqrt15, which is the perimeter.
First, we need to find the lengths of the sides. Let's do the square first. Because each angle is 90 degrees, the diagonal splits the angle into two 45 degree angles. This lets us know that there are two 45-45-90 triangles in the square. By using the ratios of a 45-45-90 triangle, we can figure out that the side length the square root is because:
sqrt12/sqrt2(sqrt2/sqrt2)=sqrt24/2=sqrt12
Next, we need to know the lengths of the triangle. Since the equilateral triangle is divided by the altitude, the altitude splits into two 30-60-90 triangles. From the 30-60-90 Triangle Theorem, we can conclude that the shorter leg is 3sqrt3 because 9/sqrt3(sqrt3/sqrt3)=9sqrt3/3=3sqrt3. Then, we need to find the length of the hypotenuse. To find this, we need to multiply 3sqrt3 by 2 to get 6sqrt3.
The last step is to add all the sides together. For the sake of easier calculations, we can simplify 6sqrt3 to sqrt108. This would give us the equation: 2(sqrt12)+2(sqrt108). Once we solve the equation, we get sqrt240 because sqrt24+sqrt216=sqrt240. When we simplify the square root, we get 4sqrt15, which is the perimeter.