Television screens are usually measured by the diagonal of the screen. The length and width of the rectangular screen become the legs of our right triangle with a diagonal that is our hypotenuse.
Find the height of each screen in the table below, round to tenth. Remember which is the legs and which is the diagonal or c. We want the TV to fit on a 17 x 50-inch shelf. What is the biggest TV (longest diagonal) that will fit on the shelf? Choose A, B, C, D
Screen
Diagonal
Width
Height
A
15”
13”
?
B
22”
19”
?
C
32”
28”
?
D
40”
35”
?
To find the height of each screen, we can use the Pythagorean theorem for right triangles, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The formula is:
c² = a² + b²
Here, c is the diagonal of the screen, a is the width, and we need to find b, which is the height.
For each screen (A, B, C, D), we'll solve for the height (b):
A. A 15-inch diagonal with a 13-inch width:
b = √(c² - a²) = √(15² - 13²) = √(225 - 169) = √56 = 7.5"
B. A 22-inch diagonal with a 19-inch width:
b = √(c² - a²) = √(22² - 19²) = √(484 - 361) = √123 ≈ 11.1"
C. A 32-inch diagonal with a 28-inch width:
b = √(c² - a²) = √(32² - 28²) = √(1024 - 784) = √240 ≈ 15.5"
D. A 40-inch diagonal with a 35-inch width:
b = √(c² - a²) = √(40² - 35²) = √(1600 - 1225) = √375 ≈ 19.4"
Rounded to one decimal place, the heights are:
A. 7.5 inches
B. 11.1 inches
C. 15.5 inches
D. 19.4 inches
To fit a TV on a 17 x 50-inch shelf, both the width and the height of the TV must be less than or equal to the dimensions of the shelf. Since we have a constraint on both the width and height, we need to check which diagonal TV size will fit within those dimensions. Let's check each TV:
A. 13-inch width and 7.5-inch height - Will fit.
B. 19-inch width and 11.1-inch height - Will fit.
C. 28-inch width and 15.5-inch height - Will not fit due to the height exceeding 17 inches.
D. 35-inch width and 19.4-inch height - Will not fit due to the width exceeding 50 inches and the height exceeding 17 inches.
From the options given, the biggest TV that will fit on the shelf with dimensions 17 x 50 inches is the TV with screen diagonal B, which is 22 inches, since options C and D either exceed the shelf height or width, and option A is smaller than option B.