Let right triangle ABC have a leg that is 9 inches long and a hypotenuse that is 41 inches long. What is the length of the other leg?
Because we have a right triangle, we can use the Pythagorean Theorem: a^2 + b^2 = c^2, for legs a and b and hypotenuse c. We have one leg that is 9 inches long and a hypotenuse that is 41 inches long. Therefore, we can set up our equation by substituting our values: (9)^2 + b^2 = (41)^2 Simplifying, we obtain: 81 + b^2 = 1681 (-81) + 81 + b^2 = 1681 + (-81) b^2 = 1600 √(b^2) = √(1600) b = 40 inches Thus, the length of the other leg of the right triangle is 40 inches.
A rectangle has a perimeter of 36 inches. If the length of one side of the rectangle is 12 inches, what is the area of the rectangle?
A rectangle is a quadrilateral that has four right angles. Because the figure has four right angles, the rectangle has a unique property that opposite sides are equal. Thus, if one side of the rectangle is 12 inches, then the other side is 12 inches. Let the width of the rectangle be x. Then, the perimeter is: 12 + 12 + x + x = 36 Adding like terms, we obtain: 24 + 2x = 36 If we subtract 24 from both sides, the equation becomes: (-24) + 24 + 2x = 36 + (-24) 2x = 12 If we divide both sides by 2, our variable, x, becomes: 2x = 12 __ __ 2 2 x = 6 Thus, our width is 6 inches. (Alternatively, the formula for the perimeter of a rectangle is 2 * (width) + 2 * (length) = Perimeter.) To obtain the area of a rectangle, you multiply the width and length: (width) * (length) = Area. Because our width is 6 inches, and because our length is 12 inches, our area is: 6 * 12 = 72 square inches. Thus, the area of the rectangle is 72 square inches.
Expand the following expression: (x + 2) * (2x - 4)
Our expression has two quantities: (x + 2), and (2x - 4). Here, we will use the FOIL method (FOIL stands for First, Outer, Inner, Last). We take the following steps to multiply the expression: 1. FIRST - Multiply the first terms in each quantity. In this example, we multiply x * 2x = 2x^2. 2. OUTER - Multiply the terms that are on the outside of the quantities. Imagine the form, (ax + b) * (cx + d). The outer leading coefficients would be a and d. Here, we multiply x * (-4) = -4x. NOTE: Notice that we multiply -4 instead of 4 because the value is being subtracted. 3. INNER - Multiply the terms that are on the inside of the quantities. Recall the previous form, (ax + b) * (cx + d). The inner leading coefficients would be b and c. Here, we multiply 2 * 2x = 4x. 4. LAST - Multiply the last terms in each quantity. Here, we multiply 2 * (-4) = -8. 5. Add the products together. After multiplying the first, outer, inner, and last terms, add the products together. Here, we have 2x^2 + (-4x) + 4x + (-8). 6. If needed, simplify. Notice that (-4x) and 4x contain the same variable, x. They can be added together, resulting in 4x + (-4x) = 4x - 4x = 0x. Thus, our expression can be reduced to 2x^2 + 0x + (-8) = 2x^2 + 0x - 8 = 2x^2 - 8. Thus, our expanded expression is 2x^2 - 8.