Vinny's Practice Problems
Question 1: A square has both diagonals drawn in it. If one side of the square is 18 feet, what is the area of one small triangle formed by the two diagonals?
Answer: 81 feet squared
Explanation: There are multiple ways to solve this problem, but here is what I did. First, I realized that all four small triangles formed by the two diagonals are 45-45-90 triangles. This means that, if the 18 is across from the right angle, then the other sides are 18/(sqrt 2), or 9 (sqrt 2). Since the triangle is a 45-45-90, the other leg is also the same length. Finally, I plugged these two lengths into the area formula for a triangle, (1/2)bh, to find the final area. The equation simplified to (1/2)(9*sqrt 2)(9*sqrt 2), which went to (1/2)(162). This is how I got 81 square feet for my final answer.
Question 2: A right triangle has legs of 24 feet and 34 feet. The hypotenuse is x feet. Using this information, what is the perimeter of the right triangle? Simplify the answer to simplest radical form.
Answer: 58+2(sqrt 433)
Explanation: The first thing to do is plug 24 and 34 into the Pythagorean Theorem for a and b, and simplify the equation. It simplifies all the way down to c = sqrt 1732. Then, this can be simplified to 2(sqrt 433). The last step is to add up all the sides to get the perimeter. 24+34 is 58, but since you cannot add this to a radical, 58+2(sqrt 433) is the final answer.
Question 1: A square has both diagonals drawn in it. If one side of the square is 18 feet, what is the area of one small triangle formed by the two diagonals?
Answer: 81 feet squared
Explanation: There are multiple ways to solve this problem, but here is what I did. First, I realized that all four small triangles formed by the two diagonals are 45-45-90 triangles. This means that, if the 18 is across from the right angle, then the other sides are 18/(sqrt 2), or 9 (sqrt 2). Since the triangle is a 45-45-90, the other leg is also the same length. Finally, I plugged these two lengths into the area formula for a triangle, (1/2)bh, to find the final area. The equation simplified to (1/2)(9*sqrt 2)(9*sqrt 2), which went to (1/2)(162). This is how I got 81 square feet for my final answer.
Question 2: A right triangle has legs of 24 feet and 34 feet. The hypotenuse is x feet. Using this information, what is the perimeter of the right triangle? Simplify the answer to simplest radical form.
Answer: 58+2(sqrt 433)
Explanation: The first thing to do is plug 24 and 34 into the Pythagorean Theorem for a and b, and simplify the equation. It simplifies all the way down to c = sqrt 1732. Then, this can be simplified to 2(sqrt 433). The last step is to add up all the sides to get the perimeter. 24+34 is 58, but since you cannot add this to a radical, 58+2(sqrt 433) is the final answer.
Dhruv's practice problems
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ANiket's Practice Problems
Question 1:
Laura has a square garden in her backyard. She wants to create an irrigation line across the diagonal. If the perimeter of the garden is 72 feet, what is the length of the irrigation line?
Answer: 18\sqrt{2}
Explanation: There are many ways to solve it but here's the simplest one. Since the perimeter is 72, the side length of the garden must be 18 feet - 72/4. As a result you must find the hypotenuse of a triangle with legs 18. 18^2+18^2=c^2. 324+324=c^2. \sqrt{648}=c, c=18\sqrt{2} Overall, after solving for the side length, and then separating the square into two 45-45-90 triangles, and solving for the hypotenuse, you find that the length of the irrigation line is 18\sqrt{2}.
Laura has a square garden in her backyard. She wants to create an irrigation line across the diagonal. If the perimeter of the garden is 72 feet, what is the length of the irrigation line?
Answer: 18\sqrt{2}
Explanation: There are many ways to solve it but here's the simplest one. Since the perimeter is 72, the side length of the garden must be 18 feet - 72/4. As a result you must find the hypotenuse of a triangle with legs 18. 18^2+18^2=c^2. 324+324=c^2. \sqrt{648}=c, c=18\sqrt{2} Overall, after solving for the side length, and then separating the square into two 45-45-90 triangles, and solving for the hypotenuse, you find that the length of the irrigation line is 18\sqrt{2}.
Question 2: A triangular award is given to James. He is trying to customize a case for this award to display it. He must fine appropriate lengths and he wants to make it a right triangle. There is a man at the very bottom right of the award and his line of vision is 30 degrees. If the one leg is 10 what is the longest side of this case and how?
Answer: 20\sqrt{3}/3
Explanation: Since all angles in a triangle equal 180 degrees, the third angle must be 60 degrees. This therefore is 30-60-90 triangle. The rules for a triangle such as this is so that the length of the longest side/hypotenuse is twice the length of the shortest leg(across from the 30°), and the length of the longer leg(across from the 60°) is the length of the shorter leg times \sqrt{3}. Therefore, the other leg would be 10\sqrt{3} which equals 10\sqrt{3}/3. Since the hypotenuse is twice this leg, the longest side or hypotenuse would be 20\sqrt{3}/3.
Answer: 20\sqrt{3}/3
Explanation: Since all angles in a triangle equal 180 degrees, the third angle must be 60 degrees. This therefore is 30-60-90 triangle. The rules for a triangle such as this is so that the length of the longest side/hypotenuse is twice the length of the shortest leg(across from the 30°), and the length of the longer leg(across from the 60°) is the length of the shorter leg times \sqrt{3}. Therefore, the other leg would be 10\sqrt{3} which equals 10\sqrt{3}/3. Since the hypotenuse is twice this leg, the longest side or hypotenuse would be 20\sqrt{3}/3.