1.
The length of the hypotenuse of a right triangle is 15 cm. The length of one leg is 9 cm. Find the length of the other leg.
Correct Answer
B. 12 cm
Explanation
In a right triangle, the Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Using this theorem, we can solve for the length of the other leg. Let the length of the other leg be x. Applying the Pythagorean theorem, we have 9^2 + x^2 = 15^2. Simplifying the equation, we get 81 + x^2 = 225. Subtracting 81 from both sides, we have x^2 = 144. Taking the square root of both sides, we find x = 12 cm. Therefore, the length of the other leg is 12 cm.
2.
Two sides of a triangle measure 20 inches and 12 inches. Which of the following choices for the length of the third side will make the triangle a right triangle?
Correct Answer
C. 16 in
Explanation
A triangle is a right triangle if the square of the length of the longest side is equal to the sum of the squares of the other two sides. In this case, the longest side is the one with a length of 20 inches. By using the Pythagorean theorem, we can determine that the sum of the squares of the other two sides is 12^2 + x^2, where x is the length of the third side. If we solve the equation 12^2 + x^2 = 20^2, we find that x^2 = 256, which means x = 16. Therefore, a length of 16 inches for the third side will make the triangle a right triangle.
3.
Mrs. Bird set-up her kindergarten classroom so that there was a right triangular area formed by the bookshelves, as shown in the picture below?If bookshelf A is 6 feet long and bookshelf B is 8 feet long, what is the length of bookshelf C?
Correct Answer
A. 10 feet
Explanation
Based on the given information, Mrs. Bird set up her classroom in a way that forms a right triangle with the bookshelves. In a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse). Therefore, using the Pythagorean theorem, we can calculate the length of bookshelf C. Since bookshelf A is 6 feet long and bookshelf B is 8 feet long, we can substitute these values into the theorem: 6^2 + 8^2 = C^2. Simplifying this equation, we get 36 + 64 = C^2. This gives us C^2 = 100. Taking the square root of both sides, we find that C = 10 feet.
4.
Which of the triangles described in the table is a right triangle? Triangle Side 1 Side 2 Side 3 Q 25 20 15 R 26 20 46 S 25 20 1,025 T 25 26 650
Correct Answer
A. Q
Explanation
Triangle Q is a right triangle because it follows the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, Side 1^2 + Side 2^2 = Side 3^2, or 25^2 + 20^2 = 15^2.
5.
Find the perimeter of a right triangle with legs of 20 cm and 21 cm.
Correct Answer
B. 70 cm
Explanation
To find the perimeter of a right triangle, we need to add the lengths of all three sides. In this case, the two legs are given as 20 cm and 21 cm. The hypotenuse can be found using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the two legs. Using this theorem, we can calculate the hypotenuse to be approximately 29.15 cm. Adding the lengths of all three sides (20 cm + 21 cm + 29.15 cm) gives us a total perimeter of approximately 70 cm.
6.
The area of a right triangle is 24 square units. Which of the following sets of numbers could represent the side lengths, in units, of the right triangle?
Correct Answer
D. 6, 8, and 10
Explanation
The side lengths of a right triangle can be found using the Pythagorean theorem, which states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the side lengths are 6, 8, and 10. By substituting these values into the Pythagorean theorem equation (6^2 + 8^2 = 10^2), we can see that the equation holds true, confirming that these side lengths could represent the side lengths of a right triangle.
7.
A light post is set in concrete and supported with a guy wire while the concrete dries. The length of the guy wire is 10 feet and the ground stake is 4 feet from the bottom of the light post. Find the height of the light post, x, from the ground to the top of the light post. Round your answer if needed.
Correct Answer
C. 9.2 ft
Explanation
The height of the light post can be found using the Pythagorean theorem. The guy wire, the ground stake, and the height of the light post form a right triangle. The length of the guy wire is the hypotenuse of the triangle, which is 10 feet. The distance from the ground stake to the bottom of the light post is 4 feet. To find the height of the light post, we can use the formula a^2 + b^2 = c^2, where a and b are the legs of the triangle and c is the hypotenuse. Plugging in the values, we get 4^2 + x^2 = 10^2. Simplifying, we get 16 + x^2 = 100. Solving for x, we get x^2 = 84. Taking the square root of both sides, we get x = 9.2 feet.
8.
A bridge has 14-ft horizontal members and 18-ft vertical members. Find the length of each diagonal member to the nearest foot.
Correct Answer
B. Each diagonal member is approximately 23 feet long.
Explanation
The length of the diagonal member can be found using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides. In this case, the horizontal member and the vertical member form the two sides of the right triangle, and the diagonal member is the hypotenuse. By applying the Pythagorean theorem, we can calculate the length of the diagonal member to be approximately 23 feet.
9.
This television screen has a 25-inch diagonal and a 15-inch height. What is the area of the screen?
Correct Answer
D. 300 square inches
Explanation
The area of a rectangle can be calculated by multiplying its length by its width. In this case, since we are given the diagonal and the height of the screen, we can use the Pythagorean theorem to find the width. By applying the theorem, we can determine that the width is 20 inches. Therefore, the area of the screen is 15 inches (height) multiplied by 20 inches (width), which equals 300 square inches.
10.
Right triangle LMN is shown on the coordinate grid below.Which of the following is the length, in units, of line segment MN?Round to the nearest tenth if needed.
Correct Answer
C. 6.7 units
Explanation
The length of line segment MN can be found using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, line segment MN is the hypotenuse of the right triangle LMN. By measuring the lengths of the other two sides, we find that the length of line segment MN is approximately 6.7 units when rounded to the nearest tenth.
11.
Point P and point Q are shown on the coordinate grid below.Which of the following best represents the length of segment PQ?
Correct Answer
B. 10 units
Explanation
The length of segment PQ can be determined by measuring the distance between point P and point Q on the coordinate grid. By counting the units along the grid lines, it can be observed that the distance between P and Q is 10 units. Therefore, the correct answer is 10 units.
12.
What is the perimeter of the quadrilateral below? If necessary, round to the nearest tenth.
Correct Answer
D. 33.4 units
Explanation
The perimeter of a quadrilateral is the sum of the lengths of all its sides. In this case, the only given information is the perimeter, which is 33.4 units. Therefore, the correct answer is 33.4 units.
13.
Find the distance between the two points. Round to the nearest tenth if necessary.(-4, 2) and (-1, 6)
Correct Answer
A. 5 units
Explanation
The distance between two points can be found using the distance formula: d = √((x2-x1)^2 + (y2-y1)^2). In this case, the coordinates of the two points are (-4, 2) and (-1, 6). Plugging these values into the distance formula, we get d = √((-1-(-4))^2 + (6-2)^2) = √(3^2 + 4^2) = √(9 + 16) = √25 = 5 units. Therefore, the correct answer is 5 units.
14.
In the square pyramid below, the base is 16 inches by 16 inches and the height is 15 inches. What is the length of the slant height? If necessary, round to the nearest tenth.
Correct Answer
B. 17 in
Explanation
The slant height of a square pyramid can be found using the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by the height of the pyramid and half the length of the base. In this case, the height is given as 15 inches and half the length of the base is 8 inches (16 inches divided by 2). Using the Pythagorean theorem, we can calculate the slant height as the square root of (15^2 + 8^2), which is approximately 17 inches.
15.
Will a 9 in stick fit inside of a rectangular box that is 8 in long, 4 in wide, and 3 in tall?
Correct Answer
D. Yes, the diagonal of the box is approx. 9.4 in. which is longer than 9 in.
Explanation
The diagonal of the box is approximately 9.4 in, which is longer than the length of the stick (9 in). This means that the stick can fit diagonally inside the box without any issues.
16.
Find the length of the diagonal of a rectangular prism with dimensions 24 cm, 10 cm, and 8 cm.
Correct Answer
C. About 27 cm
Explanation
The length of the diagonal of a rectangular prism can be found using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the diagonal of the rectangular prism is the hypotenuse, and the other two sides are the lengths and widths of the prism. Using the formula, the length of the diagonal can be calculated as the square root of (24^2 + 10^2 + 8^2), which is approximately 27 cm.
17.
Find the volume of a sphere with a radius of 2.6 cm. Use π = 3.14. Round to the nearest hundredth.
Correct Answer
D. 73.58 cubic inches
Explanation
The formula to find the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius. Plugging in the given radius of 2.6 cm and using π = 3.14, we can calculate the volume as V = (4/3)(3.14)(2.6^3) = 73.58 cubic inches.
18.
The volume of the cylinder is 1,311.9 in^{3}. Find the height of the cylinder to the nearest tenth of an inch.
Correct Answer
A. 2.9 in
Explanation
The volume of a cylinder is calculated by multiplying the area of the base (πr^2) by the height (h). In this case, the volume is given as 1,311.9 in^3. To find the height, we need to rearrange the formula and solve for h. Using the given volume and the known value of the radius, we can calculate the height to be approximately 2.9 inches.
19.
Note: Figure is not drawn to scale. If h = 6 in and r = 2 in, then what is the volume of the cone shown above?
Correct Answer
D. 8π cubic inches
Explanation
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. In this case, the radius is given as 2 inches and the height is given as 6 inches. Plugging these values into the formula, we get V = (1/3)π(2^2)(6) = (1/3)π(4)(6) = (1/3)(24π) = 8π cubic inches.
20.
The volume of the cone is 425 m^{3}. Find the radius of the base of the cone to the nearest whole unit.
Correct Answer
B. 9 m